[trykondev]

I think white just continues moving the king toward the center -- if black plays waiting moves, white can bring the king to c1-d1-e1 and then shuffle the king between e1 and f1. Eventually white will sacrifice the rook in the same pattern as the main line, and then play Ke2 -- there's no "wrong" square for the white king to be stuck on such that Ke2 would be unreachable, so black doesn't gain anything from the extra waiting moves.

[microtherion]

Ah, sorry, I had a typo, I meant 1. ... Rh8, after which the rook sacrifice does not quite work the same way anymore. I agree that 1. ... Re8 does not help.

[trykondev]

Ah, understood -- I spent some time trying to figure it out after seeing your clarification. You are right that if you try the same rook sac technique, white won't be able to get the white king over in time to defend g2.

It turns out there is another very impressive move white needs to spot: after 1...Rh8, white needs to find 2. Rf8! and now black has to either trade into the drawn king and pawn ending, or take the rook and give back the tempo that's just been saved. Now the white king is in time again. Quite a move, that should be listed as a sideline in the puzzle for sure!

[microtherion]

Brilliant! That's the move I was missing.