If there are genuine moral truths, then they are objective truths. But this is not very interesting, because all it amounts to saying is that there is an appropriate moral authority out there who says this thing that must be done.
The harder question is what to say about the rational constraints that produce the authority in the first place. Are these king-making features that ground moral authority “objective”?
Well… sort of. The funny thing about moral authority is that it gets lost when it stops being worthwhile. This is in apparent contrast to legal authority, which arguably continues onward beyond the point where it has stopped being worthwhile. Still, these are matters of degree, not of kind — morality is relatively more sensitive to the idea of worthiness than law, but both do care.
So the right thing to do is to ask, not whether or not the claims are true or false, but instead: when is the authority *earned*? Some practices will be geared towards the achievement of goals (“technology-like”), others will be more geared towards maintaining expectations (“convention-like”), and some are geared towards the expression of lived experience (“art-like”). Morality qua morality is a kind of social technology, whose central point is to get things done by telling us who has the best advice on how to get on with things. In contrast, law qua law is a set of conventional verdicts, whose central point is to keep things going as they’ve been going.
If moral authority were earned only as a means towards the expression of lived experience, then the grounds of moral authority would turn out to be subjective, in the relevant sense. In those conditions, some kind of error theory would be best to hold. But I am inclined to believe that error theory is a mistake, so I won’t be saying that.