Any reference is welcome.
I've also sketched out how one might define weak cubical omega-category in a similar style, although there's one important hole in this which I haven't been able to fill. There have probably been other attempts to get a notion of weak cubical omega-category. However, we seem not to understand weak functors very well at the moment. Batanin has a definition of weak functor for the notion of weak omega-category he presents, but I don't know that a similar thing has been done in the cubical context. Advances in Mathematics 136, pp.