After playing around with the position, it looks more complex. White h5 looks mandatory to prevent Black from closing the King side with Pawn to h5; yet this wing will nevertheless be closed by Black using its f Pawn - under the threat of the white King journey you mentionned. But then Black will lose any mobility but its King; the white King can go back to Queen side and win opposition using Pawn to c7 sacrifice. Then the clear-up happens, if I am correct.
MfG,
Eric