On day 1 the BG always protects himself. Even if he doesn't go public, he'll protect himself.
This won't be a long game. It's gonna last 2-5 days.
The Cs can't allow to lose a single day, because, like I said 3 times already, the game could be over by tomorrow night.
If I'm the D, I'd now know that of the remaining 11 players, 3 are Gs and 8 are Cs. I'd also know that I can't go wrong more than once. My day 1 investigation would be crucial. I'd love to eliminate as many suspects possible. If I know who's the BG, then its 1 less suspect.
Also, the BG goes public. Then he asks the jailer to contact him in the next 20 minutes and they decide together who to jail (the jailer has 35 minutes left to jail someone). The BG knows who's the jailer.
Then he asks the D to contact him and gives the D 3 hours to do that. If only 1 person contacts him in those 3 hours, it's likely that it's the real D. The BG tells the D who's the jailer, so now the D has one more guy to eliminate and he has 9 persons left, 1/3 of which are Gs.