Puglia 2007
As announced in his comment, AlexB got ahold of Puglia 2007, sent it to the mods, and our tricipitous mod forwarded it to me. I read it, and I can now answer that part of this question.
So, with a lot of pretty solid arguments, the article proposes the following collage of P.Oxy. 1787 fragments (where 87(13) and 87(14) have swapped numbers):

From this, we can see the following:
- The famous mysterious line endings are actually from P.Halle, the leftmost fragment in the pic, and were missed by all previous transcribers;
- The joining of P.Halle with P.Oxy. 1787 fr. 3 is exactly as I guessed in my answer to the question of this question's body;
- Thanks to the topmost fragments, P.Halle must have lost three lines at its top, whence 5 lines reside between τύχοισα and the other fragment; or rather, at least 5.
The last missing bit is: "Why exactly five, and not, say, six?". Well, let's look at the other fragment that goes before this collage:

See that little protrusion at the bottom-right? It's empty. This means the last line of the fragment was the last of its column.
Now, we may say, "Why should we assume this fragment went immediately before the P.Halle lines?". Well, we have good reasons (cfr. history in the question body) to place it before said lines, and if we suppose a full other column between the two, the result is that we have a single poem that is way too long compared to the two whose lines are all partially preserved on fr. 3 and its ends 87(13)+87(14) – and we know those have no extra lines because they are between coronides. Therefore, the lacuna length is settled.
Moreover, we can also say that, after the last line of the collage 6.A.vi here, there were 6 lost lines, the line ending in ]ΗΡΟϹ on fr. 3, and then collage 6.A.v. Who knows where the poems split from each other…
P.Oxy. 2506, as far as I can say
In the below list, TBI stands for To Be Investigated
- Fr. 1 is a mess with fragments (a)-(r) but skipping (f) for whatever reason; no idea why they are combined at all, let alone the way they are; [TBI]
- Fr. 2 only reads Ο .[ | Π̣, so why is it not Sappho?
- Fr. 3 is merely ]. Α . .[ | ]ΙΝΤ[, the I looking like a ि, so why is it not Sappho?
- Fr. 4 contains the beginnings of 15 lines [TBI];
- Fr. 5 is actually 3 fragments, (a)-(c), no idea why they are combined; [TBI]
- Fr. 6 is ditto; [TBI]
- Fr. 7 reads ] . . Λ̣C̣Ọ[ | ]C̣ỊΥΓ€Ρ[ | ]Χ€ΤΙΔΟ[ | ]ΡΟΝΛ€Γ . [ | ]ΤΑΥΤΗ[ | ]Δ€Ν€Ϲ̣[; [TBI]
- For fr. 8, I agree with the Voigt transcription, pointing out that the Y ending l. 4 and the C starting l. 5 are initially completely unreadable and only identified through context, which I denote with underlines; in particular, that letter in l. 4 is identified thanks to reading the name Γογγύ[λ- in the fragment, which is also why this is Sappho;
- Fr. 9 reads ] . ΤΑΥ̣[ | ]. ΟΙϹ[ | ]ωΙϹ[ | ].[; [TBI]
- Fr. 10 reads ] . [ | ]. ϹΟ .[ | ]ωΦ̣[ | ]ΚΡΑ[, could l. 3 read σ]ώφ[ρων, as in Sophron? [TBI]
- Fr. 11 reads ]ΔΙϹ̣[ | ]ΙΔΙ[ | ]Υ̣Ϲ[; [TBI]
- Fr. 12 reads ]ωΧ̣ | ]ΙΟ̣C̣; [TBI]
- Fr. 13 is two fragments, (a) and (b), lined up vertically in the picture;
- Fr. 14 ]. .[ | ]. ΚΟ̣[ | ]€ϹṬ[ | ]Υ̣Ị[; [TBI]
- Fr. 15 reads ].[~5 letters].[ | ]Γ̣ΑΡΕΥ̣ . Ϲ€[ | ]€ΚΡΟΝ . [; [TBI]
- Fr. 16 is ]ΝΟΥ[; [TBI]
- Fr. 17 contains portions of 9 lines; [TBI] [end of pic 1]
- Fr. 18 appears to be two fragments, (a) and (b); I say "appears" because there is no number; [TBI]
- Fr. 19 reads ] . ΞΑ[ | ]ΟΝΤ[, and is another unnumbered fragment; [TBI]
- Fr. 20 reads either ]ω€ or ]ωΘ; [TBI]
- Fr. 21 reads ] . .[. .].[ | ]ỊϹΑΝΑ[ | ]€̣ϹΤΑΤ€[; [TBI]
- Fr. 22 contains the ends of 6 lines; [TBI]
- Fr. 23 reads €̣N[ | Δ€[ | Τ[; [TBI]
- Fr. 24 reads ]. ΑΙΝ .[ | ]€ΙΛΟ[; [TBI]
- Fr. 25 contains the ends of 8 lines; [TBI]
- Fr. 26 consists of fragments (a)-(f), of which (c) seems to consist of two fragments, and (e) of 3; 26(a) is assigned to Stesichorus, and I have yet to understand why; [TBI] [end of pic 2]
- Frr. 27-29 are microscopic; [TBI]
- Fr. 30 contains the beginings of 8 lines;
- Frr. 31-41 are microscopic; [TBI]
- Fr. 42 consists of fragments (a)-(d); concerning (a), I agree with the Voigt transcription, except that at l. 1 it's Κ€ΔΟΝΚ€[, not like Voigt's Κ€ΔΟΝ€[), l. 3 €̱[, l. 7 Χ̱ΑΡΑΞ̱[, a possible Charaxus mention hence this is Sappho, l. 8 the papyrus has no breathing (i.e. the breathing is added by Voigt because hers is not a raw transcription). (b)-(d) [TBI]
- Fr. 43 contains the ends of 3 lines of one column and the beginnings of 11 of another; col. i reads ΚΑ/ΧΑ | Α̣Λ | . Ϲ, maybe l. 1 is again Charaxus, I'll say Sappho for this one too; as for col. ii, I agree with Voigt, except that in l. 2 there is no vestige at the start, it possibly reads ΧΑΡ]ΑΞΟ[ hence Sappho, at l. 10 ]ṮH̱, at l. 11 the initial epsilon is certain because the middle stroke is visible;
- Fr. 44 is a bigger two-column fragment; for col. ii I agree with Voigt, noting there are no accents or breathing; it's ascribed to Sappho because she's mentioned in l. 1, and her brother possibly is in l. 2; as for col. i, will transcribe [TBI];
- Fr. 45 I agree with Voigt, with notes
l. 2 poss. IC or IE or ITH, l. 7 fort. M[?, l. 8 that's not g[ in Voigt but n[ (looks like g because of underdot), l. 9 ]e because middle and top, l. 20 ]. eg .[, l. 23 poss. ]HC, possible Charaxus mentions
;
- Fr. 46 contains the ends of 6 lines; [TBI]
- Fr. 47 has a Sappho mention, I agree with Voigt except for reading l. 4 as ϹΑΠΦ .[ and l. 5 as ω̣ΡΑΝΤ̣ .[; [end of pic 3]
- Fr. 48 appears (unnumbered) to consist of fragments (a)-(f); [TBI]
- Frr. 49-58 are micro; 50 and 52 are joined for some reason; [TBI]
- Fr. 59 has a mention of Sappho in l. 1, poss. ]H, then last two lines have no starting vestige (contrary to what Voigt says);
- Fr. 60 is Sappho because poss. Andro]medan in l. 2, which is the last one;
- Frr. 61-69 are micro; [TBI]
- Fr. 70 contains the starts of 10 lines; [TBI]
- Fr. 71 has 7 lines; [TBI]
- Frr. 72-76 are microscopic; [TBI]
- Fr. 77 is presumably the big fat scrap on the right edge of pic 4; [TBI] [end of pic 4]
- [Frr. 78-89 to be described];
- Fr. 90 contains the word Πίθω[νι, and thus belongs in Epicharmus's
"The Monkey", as identified by Finglass;
- [Frr. 91-107 to be described, end of pic 5];
- [Frr. 108-130 to be described];
- Fr. 131 is ascribed to Alcman by Stefano Vecchiato's Una nuova testimonianza su Alcmane in ‘P.Oxy.’ XXIX 2506, fr. 131?, given that l. 1 probably reads Αλκμα[ν;
- [Frr. 132-158 to be described].
P.Oxy. 2637
Well, I mentioned it in the comments, so I might as well delve into it. Now, this is a group of papyrus fragments, whose image can be found here, which was mostly attributed to Ibycus. Now, I decided not to go too deep into this, but the gist is that there are good papyrological / content arguments to place most of these fragments into one or more scrolls of commentaries on Ibycus. There are, however, "three" exceptions which are attributed to Sappho, namely the three below.

This is fragment 27. I read it as:
]. Ι̣ϹΑΠ
]Χ̣ΑΝΟΝ
]ΑΙϹΗ
]ΕΒΡΟ
]ΙΝΟϹ
].
The X in l. 2 may be a Z, and the vestige in the last line may just be something funny going on with the papyrus rather than actual ink (a "papyrus quirk", as I say). This is given to Sappho because l. 1 ends in ϹΑΠ[Φ-. Note that this fragments is confusingly joined to the Ibycean fr. 12(b) in the picture, and I believe the split happens either where the seemingly wannabe continuous ink stroke at the top-left breaks, or where the digitiform protuberance starts.

This is fr. 35. I think you can see why I said "three" exceptions above: this very clearly appears to be more than one fragment, plus another one below them all, which is fr. 4. Here are the borders I think I can trace:

I read the leftmost fragment as:
]+.+ .[
]. Ψ̣ .[
]Η[
]Μ[
]Κ .[
with my usual convention of using + to denote blank spaces on the papyrus.
The top-left appears to be:
]+. . .+ . Π̣[
[blank line]
The top-right fragment, the smallest of the bunch, reads:
]'ΓΗ[
]+.+[
And the biggest fragment of all reads:
]+.+[
]. .[
]+.+ . . +.+[
]Η̣ . +.+[
]Π€ΡΙΓΟΓΓ̣[
]Η̣ΛωΙ . ΒΑ̣Λ̣[ . ]Ρ̣[
]Α̣ΙΗϹΑΠΦω[
] . ΜΟΤΑΥ̣Ρ[
] . Φ̣[ . ]€ΝΑϹ€ . [
] . €ΛΛ[
]ΙΟΠΗ̣Ϲ[
] . ΠΑΘ€ΙΤ[
]Θ̣Η+.+Τ[
The following is the combined transcription:
]+. . .+ . Π̣'ΓΗ+.+[
]+. . . . . .+ . .[
]+.+ . +.+ . . +.+[
]. Ψ̣ .[ . ]Η̣ . +.+[
]Η[]Π€ΡΙΓΟΓΓ̣[
]Μ[]Η̣ΛωΙ . ΒΑ̣Λ̣[ . ]Ρ̣[
]ΚΑ̣ΙΗϹΑΠΦω[**
] . ΜΟΤΑΥ̣Ρ[
] . Φ̣[ . ]€ΝΑϹ€ . [
] . €ΛΛ[
]ΙΟΠΗ̣Ϲ[
] . ΠΑΘ€ΙΤ[
]Θ̣Η+.+Τ[
On line **, the alpha is the combination of the uncertain A in the biggest fragment, and the vestige on the leftmost one.
Well, there is one problem left here. Ll. 1-2 of this combination in my transcription appear to be usually taken as one line. My problem with that, and the reason I don't read π(ερὶ) γήρ[ως like Cambpell, is that the eta, were I to do so, would acquire a spurious little curve on its bottom-left. Here are the letters as I see them:

As you can see, the ?? doesn't belong in an eta. So I wonder: can we explain it away as a "papyrus quirk", and read the single line as Π̣'ΓΗΡ̣[ combining all fragments, or can we not?

Finally, this is fr. 38, which I read as:
].[.]. ϹΤΟϹ[
]ΑΠΟΜΥΤ[
]ΟΝϹΦ .[
L. 2 completes to ἀπὸ Μυτ[ιλήνης, hence the attribution to Sappho.
UPDATE
z-lib, olim [b-ok olim bookzz], has the Oxy volumes for both 2506 and 2637. I will dive into 2506 at some point, but now let's deal with 2637.
Fr. 27
Fr. 27 is transcribed as below:

After better inspecting the papyrus, I agree with everything. On l. 1, I note that my I is actually curved, or «the right-hand arc of a circle», as the volume puts it, and that my vestige may not be in the papyrus fragment, since the division from another scrap is doubtful.
Fr. 35
On fr. 35, they say nothing about my problem, and just smash the lines together. I'm not 100% convinced that the spurious thing is a papyrus quirk, but I guess for the sake of consistency with "e.p." I will accept this explanation.
Now, my combined transcription with fragment separations is:
]+. . .+ . Π̣[]'ΓΗ[]+.+[
]+. . . . . [] .+[]. .[
]+.+ .[]+.+ . . +.+[
]. Ψ̣ .[]Η̣ . +.+[
]Η[]Π€ΡΙΓΟΓΓ̣[
]Μ[]Η̣ΛωΙ . ΒΑ̣Λ̣[ . ]Ρ̣[
]Κ .[∠]\ΙΗϹΑΠΦω[ .[∠]\ = Ạ
] . ΜΟΤΑΥ̣Ρ[
] . Φ̣[ . ]€ΝΑϹ€ . [
] . €ΛΛ[
]ΙΟΠΗ̣Ϲ[
] . ΠΑΘ€ΙΤ[
]Θ̣Η+.+Τ[
The volume gives:

Differences:
- I have quite a few vestiges they don't; not sure why they treat them as if they weren't there; maybe it's the pic? Or maybe they are convincingly explained away as quirks? Perhaps this header thing justifies treating a few lines below it as blank?
- On l. 2 (counted from the Gongyla line), between I and B I see something on the bottom of the line, some kind of circle; punctuation mark? And the two vestiges at the end, I see connected by what is possibly a papyrus quirk, hence the doubtfulness of my rho;
- On l. 3, I guess we cannot be sure if the C closes or not, or if the middle stroke is present, making it possibly O or €; volume only sees top, I guess the bottom I'm seeing is a quirk?
- On l. 4, my initial M could be split into .Ạ, and my vestige could be a quirk (not sure where exactly I saw it); I have to say my M could be a bit too wide and slanted;
- On l. 5, I defend my transcription; «only the overhang» of their C̣ is visible, and they see «short stretches of the shank» of a rho, but they're more likely quirks than those in the heading AFAICT;
- On l. 6, I did miss a vestige, and my two lambdas do join and form a possible mu; in fact, they don't look at all like the lambda in μήλωι up in l. 2, so I'll follow the volume;
- On l. 7, I defend my certain letters, but did indeed miss a vestige at line beginning;
- On l. 9, I can see where the theta came from, and also where the volume's EI comes from; given the top doesn't close, I'll go for ẸỊ there; and of course my +.+ equates to their blank space.
Fr. 38
They read:

I read:
].[.]. ϹΤΟϹ[
]ΑΠΟΜΥΤ[
]ΟΝϹΦ .[
- On l. 3, my nu was utter bullshit, it's definitely an Y;
- On l. 2, the T is certain to me;
- On l. 1, besides my hole which is arguable, what about CTOC vs. ϹΠ€̣? Well, the sigma is unquestionable; my final C should probably be doubtful given we only have a small portion of the curve; as for TO vs. Π, well, the right leg of the alleged pi closes in on itself, so I read that as an omicron, and the rest had to be a tau; well actually, the closure is incomplete, so the omicron is impossible, and we'd get CTC only from trying to read CT there, which is impossible, I think, so I'll follow them.