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!