Checkmate with Two Queens

I challenged a group of students to find a position with two queens against a lone king that took me more than three moves to deliver checkmate. After several failed efforts, they came up with the key idea: place White's king where it interferes with the mobility of both queens. I could not find checkmate in less than four moves. Checking with an engine verifies that it is indeed checkmate in four.

White to move

Does a position exist where more than four moves are needed?