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I don’t see anything interesting for black after 1.Kf4.
For instance 1.-e5+ 2.Kf5 e6+ 3.Kf6 Kd7 4.Kxe5 Ke7 5.d4 Kd7 6.Kf6 Kd6 7.e5+ Kd7 8.Kf7
White king moves to kingside via Kf4, and black has nowhere to go.
If e5+, then Kf5, and black has to move away from e5 (whether immediately or a move after e6+). White then gobbles the e5 pawn and black is helpless.
1. Kf4 Kc5
2. Ke5 Kb4
etc. I see various lines leading to all pawns being captured (draw). I don’t see a white win but who knows.
I first try to find key squares… but that is not necessary here: Black’s king is in ZZ after 1.Kf4 and this is enough:
1.Kf4 and black is in ZZ.
1…e5 2.Kf5 ZZ (if king move pawn is lost)
2…e6+ Kf6 ZZ the king must now move, no more sparing move, e5 pawn is lost and then e6 too.
Have a nice day
Yes
1. Kf4 Kc5
2. Ke5 Kb4
3. Kxe6 Kc3
4. Kxe7 Kxd3
5. e5
is the refutation of the other line presented