Only if you cut it too big, and even then, the gearboxes will only rotate by a couple of degrees. Here's what you're effectively doing:
drill.png
So instead of cutting it to perfectly mate with the drill, the four corners (blue bits) are now just gaps. The properties of a rectanlge mean that it's not possible for any rotation to occur because the top and bottom are restrained in a gap of 25mm and the left and right sides are restrained in a 30mm gap

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