Let $\text{Cat}$ be the category of (small) categories and functors. There is a "categorical" (also called "canonical" or "folk") model structure on $\text{Cat}$ in which the weak equivalences are the usual equivalences of categories (and as explained here, it turns out that there is a *unique* choice of cofibrations and fibrations that come with them). In this model structure, all categories are fibrant and cofibrant.

There is also a natural question of constructing a "geometric" model structure on $\text{Cat}$ for which the weak equivalences are functros that induce weak equivalence of nerves. i.e. there is the nerve functor $N:\text{Cat}\to \text{sSet}$, where $\text{sSet}$ is the category of simplicial sets with the standard model structure, and we would like a model structure on $\text{Cat}$, for which a functor $F:C\to D$ is a weak equivalence if and only if $N(F):N(C)\to N(D)$ is a weak equivalence. There is such a construction by Thomason, in which the model structure on $\text{Cat}$ is "transported" from $\text{sSet}$ along the adjunction $$ \tau_1 \circ Sd^2 :\text{sSet}\leftrightarrows\text{Cat}: Ex^2\circ N $$ Where $\tau_1$ is the left adjoint of $N$ (sometimes called "fundamental category") and $Sd\dashv Ex$ are the barycentric subdivision / Extension adjunction of Kan. In Thomason's model structure, the cofibrations are quite complicated (they are all "Dwyer maps"). In particular a cofibrant category must be a poset.

My question is whether there exist a model structure on $\text{Cat}$, with "geometric" weak equivalences as in Thomason's, but with the same cofibrations as the "categorical" one. In other words, can we obtain a "geometric" model structure on $\text{Cat}$ by localizing the categorical one?