The last one came in the following position.
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I replied 38...a6?!, but 39.axb5 meant that, whichever way I recaptured, I was very unlikely to get a passed pawn.
Indeed after 39...axb5 40.Kf4 Bf7 41.h6 gxh6 42.Bxh6 Kc6 43.Bf8 the game is completely equal, according to Stockfish16 and Komodo14.1, and a draw was quickly agreed.
The alternative is 38...bxa4, after which Black admittedly has doubled and isolated pawns on the a file, but the passed a4 pawn is dangerous, eg 39.Rb7+ Kc8 40.Rxg7? a3 41.h6 Bf5 42.Bc1 a2 43.Bb2 Rb5 43.h7 Bxh7 44.Rxh7 a5! wins.
Better for White is 39.Rb2, after which the engines reckon both 39...Bf5 and 39...Ra6 give Black the upper hand. It is not clear whether Black is winning, but it is clear 38...bxa4 is the only winning attempt.
LESSON: a tidy pawn-formation is not better than an untidy won if the former only draws and the latter may win. In other words, concrete analysis is what counts, not aesthetics.
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