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Unfortunately no. After 1. Ke2 g2 2. Qe1 Black has a simple way to win the queen, and after 2. Qc1 Black can at at the very least promote the pawn and there will never be a checkmate for White.
Does it involve Ke2, Pg2, Qe1? Ending with Qc1, Qh6
Edit nvm I’m dumb
Unfortunately no. After 1. Ke2 g2 2. Qe1 Black has a simple way to win the queen, and after 2. Qc1 Black can at at the very least promote the pawn and there will never be a checkmate for White.