The question here is; does anyone who believes in Universal Morality really let it sway their own lives and behavior. Since it is clear that people have vastly different views on what is right or wrong, most people's personal convictions will be wrong (i.e. contradict with the Universal view of "right and wrong").
This means that most, if not all, people will have moral convictions that are "wrong". Anyone interested in being a Moral person should be looking for ways they are wrong to change their moral convictions, right?
I think this is fair question; if there is a Universal Morality, and you find out that something you now find intolerable (i.e. marital rape or abortion) is acceptable, or if you find out something you now find acceptable (interracial marriage) is intolerable, what do you do?
If Absolute Morality is just about everyone doing their hardest to prove that their views are the Universal views, and no one changes their views to conform with Universal morality...
... then what is the point?