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

Why would a strict social conservative consent to a contract that allowed me to have sex with multiple partners at the same time?

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 don't think many social conservatives really believe that sex is a community matter. They're just applying a good old fashioned double standard. Call them on their own sexual behaviour and they'll rush back to consensual ethics ("none of your business!") so fast you'll see Lorentz contraction.