I concur with Morendil that Rawls' "Veil of Ignorance" is a rather elegant way of showing morality to be conditionally objective.

I don't know what work "conditionally" is doing here. But I'm pretty sure Rawls himself doesn't take his theory to justice to show that morality is objective. In fact, in A Restatement he explicitly disclaims that he has demonstrated morality is objective. What he is doing is trying to formalize Western/liberal intuitions about justice.

(EDIT: Just checked. The correct interpretation of the initial publication of A Theory of Justice is that Rawls is trying to demonstrate the objective truth of liberalism, but in later publications he changes his mind in response to criticisms and agrees that he is really just formalizing this intuition of justice as fairness)

I think you may be overestimating the consistency of the social conservative viewpoint. If you were to tell them about how, when, where & why they could have sex, they would be outraged - even if you couched it in, say, biblical terms.

I'm certain there are non-hypocritical social conservatives somewhere. I don't think prohibiting polyamory while also allowing measures of sexual privacy are necessarily inconsistent. Holding that some aspect of sexual behavior should be community matters does not require holding that all aspects of sexual behavior must be community matters.