White plays and wins in 5 moves https://lichess.org/gaPDvLQD
- Kf7 g6 2. hxg6 h5 3. g7+ Kh7 4. g8=Q+ Kh6 5. Qg6#
The analysis board does checkmate in 5 moves
- Kf7 Kh7 2. Ba4 Kh8 3. Bc2 g6 4. hxg6 h5 5. g7#
I didn't need to use my bishop ;)
Source:
This publication is in my collection "500 chess puzzles. Mate in 5. Advanced & Expert Level."