A Private Conference is so that Buttock’s lawyers can say -
hey, look, heres the BC, it says he’s born in Hawaii, but it also says “religion: Muslim” or “Father: [not who he claims]”
and this would be too embarassing to bring out in public.
The upshot is, we’ll probably NEVER see it, and every treaty, law, and EO will be suspect as being invalid.
Oh, lord, no, not another millenia of “if the priest be false” arguments ~ !!!!!!!!!!!!!!!!!
A private conference is just that, a private conference among the justices themselves. Neither side will have representatives in the room.
But say, for the sake of arguement, you're correct and that the birth certificate gives a different father or states that the religion is Muslim then so what? The only Constitutional question is whether Obama is a natural born U.S. citizen. If he is then that's the end of the story, and whatever information is on the birth certificate is completely irrelevant.