I can't see if I has completed a map in DD mode so this is my idea. You get only the star if you complete it in DD mode.
We have already mentioned that somewhere. somehow, DD mode should be differentiated from normal mode. As it stands everything is lumped together, which reduces the incentive to play DD mode.
But this late in the game's life, I think it's pretty much not going to happen. Look for it as implemented in the next part that may come out sometime in the future.