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.

nottrue. $\endgroup$ – Mike Shulman Oct 14 '16 at 3:56