There's no escape for the white king here.
Bulldog Endgame Theory

Black does need to be careful of stalemate, but it would be a blunder.
With black to move here...
1... Gj4 2. Kj2 Kh2 3. Kj1 Gi4 4. Kj2 Ki3+ 5. Kj1 Gi2#
There was a tempo wasted by black there, which avoided the stalemate.
No.