I suppose, though I see no need to add the restriction "non-utilitarian".
Yeah, so the QRI theory of consciousness would probably say that moral views are primarily based on aesthetics, where an aesthetic is something like a tendency of the brain to translate a certain class of concepts to high-valence qualia and others to low-valence qualia. With respect to art, this would mean that some kinds of art styles are pleasant to look at and others aren't, but it equally applies to other contexts. E.g., one could have an aesthetic of minimalism (to take an example from myself) and generally find it pleasant to be in an environment where every distinct object has a purpose, whereas a lot of unnecessary stuff would feel unpleasant. Or one could have a nature-y aesthetic and find artificial looking things ugly and freely growing things beautiful. Or an aesthetic that looks at everything in terms of self-determination. Or a goth aesthetic. Etc. To a significant extent this concept is in the mainstream, it's also often called "vibe" or "identity" or something, so the QRI version is mostly that plus the claim that what ultimately drives the observable behavior is which mental objects feel pleasant vs. unpleasant. So it's again all about valence~
So then my guess would be that moral views are largely upstream of that. Like if you're a conservative who is all about human decency and dignity, you're probably gonna have moral views that revolve around the sanctity of life. If you're more hardcore and like strict laws, then a deontological morality would probably make the most sense. But rarely is nihilism going to fit the aesthetic.
If this is on the right track, then the most frequently rediscovered moral principles would just align with the most common aesthetics. (Worth noting here that aesthetics can definitely be environmentally shaped.) So it ultimately depends on how you carve up the world, what type of thing a moral value or rule is in your book, and how it relates to other concepts. In general, virtue ethics is probably the kind of thing that most people find attractive, and then utilitarianism might be a little more common in analytically minded cultures.
As far as evidence for this elaborate theory goes, the experiment would be to think of a moral principle you subscribe to and then just introspect on how you feel about that principle, not intellectually but emotionally. Like, what is the vibe of the qualia that represents the concept? Then think of a principle that you don't like and again look at the vibe. The model predicts that the one you like feels much higher valence. (Ofc you could argue causality goes the other way here.) Ironically this could mean that the concept of valence feels low-valence, which even if it's not true for you is probably true for lots of people. Whereas the concept of dignity could feel really high valence. I think I've talked about this general idea before.
Always worth mentioning here that valence is not pleasure or any other particular form of positive emotion; it's maximally broad and encompasses any and all forms of liking.