First there is a mate in 27 moves:

1… Kf7 2.Sg4! Kg6 3.Ke7! Kf5 4.Sf2! Kg6 5.Ke6! Kg7

(5… Kh6 6.Kf6! Kh5/Kh7 7.Kf7! -  similarly)

6.Sg4! Kg8 7.Ke7! Kg7 8.Sge5! Kh7 9.Kf7! Kh6

10.Kf6! Kh5 11.Kf5! Kh6 12.Sd7! Kg7 13.Ke6! Kg8 14.Ke7! Kh8 15.Kf7! Kh7 16.Sf8+ Kh6 17.Kf6! Kh5 18.Kf5! Kh6 19.Se6! Kh5 20.Sg7+! Kh6 21.Kf6! Kh7 22.Sf5! Kg8 23.Ke7! Kh7 24.Kf7! Kh8 25.Sg5! f3

26.Se7/Sh4 f2 27.Sg6#

1.Se4? Kf7!

In the solution different plan:

1.Sg4! Kf7 2.Sfe5+!! Kf8 3.Kd8!! Kg8 4.Ke8! Kg7

5.Ke7! f3 6.Sf2! Kh7 7.Kf7! Kh6 8.Kf6! Kh7

9.Sd7! Kh6 10.Sf8! Kh5 11.Sg6! Kh6 12.Sf4! Kh7

13.Se6! Kh6 14.Sg7! Kh7 15.Sf5! Kg8 16.Ke7! Kh7

17.Kf7! Kh8 18.Kg6! Kg8 19.Sg7! Kf8 20.Kf6! Kg8

21.Se6! Kh7 22.Kg5! Kg8 23.Kg6! Kh8 24.Kf7! Kh7

25.Sg4! f2! 26.Sf8+! Kh8 27.Sf6! f1Q 28.Sg6#

(3… f3 4.Sf2! Kg8 5.Ke8! Kg7 6.Ke7! – similarly)