In Verity's first paper on weak complicial sets, he shows that every strict complicial set is a weak complicial set. He also showed in an earlier paper that the full subcategory of stratified simplicial sets spanned by the strict complicial sets is equivalent to the category of globular strict $\omega$-categories.


A troubling corollary of these two facts together is that upon passing to the homotopy category, since all strict complicial sets are both fibrant and cofibrant, that all pseudofunctors between strict $\omega$-categories are representable by strict $\omega$-functors. 


Is there a general coherence result for strict $n$-categories showing that indeed every pseudofunctor can be strictified up to a sufficiently weak notion of pseudonatural homotopy? (I am defining a pseudofunctor $A\to B$ of strict $n$-categories following Garner to be a strict $n$-functor $C(A)\to B$ where $C$ is the cofibrant replacement comonad arising from the Free-Forgetful adjunction to $n$-computads).


If not, has anyone conjectured a class of monomorphisms of stratified simplicial sets at which we could localize in order to obtain the correct model category?



Weak complicial sets: Are the morphisms too strict?
Indeed there is no such coherence result: it is false already for $2$-categories (see for instance Lemma 2 of [this paper of Steve Lack](http://tac.mta.ca/tac/volumes/18/1/18-01abs.html)). The solution to your troubling corollary is that the "correct" models for weak $\omega$-categories are not the weak complicial sets but the **saturated** weak complicial sets, i.e. those weak complicial sets in which all the equivalences are marked. In the Street nerve of a strict $\omega$-category only the identities are marked, and it is thus not saturated in general (unless the $\omega$-category has no non-identity equivalences, in which case all pseudofunctors into it are strict).


The saturated weak complicial sets are indeed the fibrant objects of a model structure on the category of stratified simplicial sets, which is a localisation of the model structure whose fibrant objects are the weak complicial sets. See Emily Riehl's lecture notes [Complicial sets, an overture](https://arxiv.org/abs/1610.06801), in particular Example 3.3.5.


