Yes nice study, BTW. Had fun with it.

1. de! is necessary, to prevent 1.__e4 Once Black has a protected passer, it's over. After 1. __ Ke5, then White must play 2. Kg3 to get some counterplay going. Passive action would be death. IMO, the correct continuation follows:
2. -- Ke4! 3. Kf2! h5! 4. e2 d4 5. Ke2 d4 5; g3 d3+ 6. Kd2 Kd4 7. g4! draw

A nice study. White can't hold this by playing 1.dxe5 and then 2.Kxg3 and then advancing his kingside majority because black will queen at d1 in a total of 8 moves, while it would take white 10 moves.

I am not quite sure right now what white has to do. I think white will also lose with 1.Kxg3 because black can foil this with 1.....h5!! 2.Kf3 e4!!

Right now, I am leaning towards 1.h5, but am still having trouble figuring out what to do after that. I think white can't win this, so is playing for a draw.