Unplayable turn in Beethoven sonata

Question

Beethoven's piano sonata Op. 2 No. 3, first movement, bars 179-180 (Henle edition).

The turn decorating each eighth note can be played literally. Except for the one on bar 179's A sharp, because there's no good candidate for that turn's lower auxiliary note. G natural is too low. G sharp is harmonically bogus. G double sharp is Chopin, not Beethoven.

So should the B - A sharp with turn be instead simplified into a B with turn, where A sharp is the lower auxiliary?

Should the dozen similar instances also be simplified thus?

I prefer arguments stronger than so-and-so recorded it like such-and-such.

Edit: A turn with a tiny sharp underneath occurs in bar 27 and in bar 161, both in the Henle and in the first edition (p. 32 bottom, p. 36 middle; although here the sharp is above, not underneath). Had Beethoven wanted the Casella interpretation, he would have notated it that way too: a quarter note B with a turn plus sharp. So we can't explain the Casella as a modernization of an archaic spelling.

Answer 1

Take a look at the first edition: http://conquest.imslp.info/files/imglnks/usimg/b/b1/IMSLP51972-PMLP01414-Beethoven_-_Piano_Sonata_No.3_(Artaria).pdf

Inverted symbol but of course it's not an inverted turn.

Now take a look at edition Casella 1919 http://conquest.imslp.info/files/imglnks/usimg/8/81/IMSLP68708-PMLP01414-Opus_2_no_3.pdf The figure is to be played as noted here.

Answer 2

Shai Wosner takes the turns as given by Casella. But Paavali Jumppanen treats each turn as if the note it's with is the turn's principal note. He plays G double sharp.

Answer 3

Beethoven goofed. It's up to individual taste whether to play Casella's triplets or literal turns. But at a fast enough tempo and with some pedal, the unbeethovenian G double sharp can be tolerably overlooked in the resulting flurry.