Mirror Chess

Bobby and Garry are playing chess. If Garry is playing with the black figures and agrees to mirror every move Bobby makes, what is the minimum number of moves in which Bobby can mate his opponent?

Bobby can mate Garry in four moves:

1. d4 d5 2. Qd3 Qd6 3. Qh3 Qh6 4. Qxc8

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