$\Theta$-Sets and Higher-QuasiCategories In his well-known paper "Disks, Duality and $\Theta$-Categories, Joyal defines at the very end a $\Theta$-Category to be a cellular set suitably "fibrant".
I was wondering whether someone has worked on this proposed model for weak $\omega$-Category, and its relationship with Dimitri Ara's Higher-QuasiCategories and Harry Gindi's attempt of putting a model structure on cellular sets.
A question could be, for instance: is it the case that the localizer for higher-quasicategories produces the same fibrant objects, namely those with the inner-horns filling property?
Any idea or comment is welcome!
 A: I think my preprint answers this question in the appendices when I compare the horizontal Joyal model structure on $\Theta[C]$ sets with the Rezk model model structure for $\operatorname{Se}_C\cup \operatorname{Cpt}_C$-local Reedy-fibrant simplicial presheaves.
If you unwind what the vertical model structure localizer is, you can extract from this that all of the Oury inner horn inclusions are local in Ara's model structure (by induction on $C=\Theta_n$). It's somewhat easy to see that you still need the higher dimensional "completeness maps".
Yuki Maehara said he has full results showing that even "bigger" horns than the Oury horns generate the inner anodynes as well, specifically the horns suggested by Joyal and Berger (omitting exactly one codimension 1 subobject), but I don't think he has made his preprint available yet.
A more useful heuristic argument to see why Joyal's proposed model structure doesn't work is that since all strict $\omega$-categories are fibrant in that model structure (they all admit unique inner horn fillers) and the embedding of strict $\omega$-categories is full in cellular sets, it would mean that every pseudofunctor between strict $\omega$-categories is naturally (not pseudonaturally) isomorphic to a strict one, which is trivially false.
Edit: Here is Yuki's paper, which came out a bit after I gave this answer.  It only covers the case n=2, but I can't imagine that this is too hard to generalize to higher n.
