On reflection, I disagree that consciousness seems to supervene on 3D states. Consciousness seems to me to be a necessarily temporally extended phenomenon. I don't know what it would mean for an object that only existed for a single instant to be conscious.

Consider this thought experiment: Suppose you were able to freeze me right now, so that all processes in my body halted. In particular, my brain remained frozen in its current state, with nothing happening in it. Would this frozen version of me be experiencing things continuously? It seems to me that both science and intuition strongly suggest that the answer is "no", that the frozen me would be unconscious. Yet if you believe that consciousness supervenes on 3-D states, then you would have to say that the answer is "yes". Each 3-D slice of that frozen person is identical to my current 3-D slice (or the 3-D slice of me a few moments ago, to be more accurate), and there's no denying that I am currently experiencing things. If those experiences supervene on my 3-D state, then they must be shared by the frozen version of me, but that seems pretty counter-intuitive to me.

So in a sense I agree that consciousness is a property of 4-D entities. Not the entire 4-D entity corresponding to a person's space-time worm, but of thin (but not infinitesimally thin) slices of that worm.

Also when you say that empirical entities seem to be 3-D, you are presumably talking about prima facie appearance. Our most sophisticated empirical understanding of the world suggests that those entities are 4-D, since presentism (and even endurantism) has huge problems reconciling itself with relativity. It would be a poor empiricist who placed more credence in prima facie empirical evidence than the carefully refined empirical evidence provided by our best science.