If your engine fails in a DA-40, you've lost oil pressure, so there likely isn't enough to push the blade to the coarse position.
A DA-40 doesn't have an accumulator like a Seminole, for example. So when it restarts, the start position is already a fine pitch before cranking the engine, and the oil pressure allows the blade to be pushed to the coarse position, not the fine position, which is also the reverse of my exemplary Seminole, which needs the oil pressure to maintain a fine pitch, so the Seminole's shutdown default is course for the same reason the DA-40's is fine: a lack of oil pressure.
The reason the emergency procedures don't specify is because the system isn't designed to allow you to feather (not that a DA-40 can feather at all) or even coarse the propeller without the oil pressure generated by the engine, and you wouldn't have much oil pressure if the engine failed.
At best, the windmilling propeller would turn the oil pump, which would provide some oil pressure, which would then coarsen the propeller to some degree, but as soon as the propeller RPM drops further, so would oil pressure with it and that would lead to the propeller going toward a more fine pitch again.