The Joyal model structure on the category of simplicial sets, has monomorphisms as cofibrations and quasi-categories as fibrant objects (these model $(\infty,1)$-categories). In HTT (section 2.3.4) Lurie defines the notion of an $n$-category (these model $(n,1)$-categories) and proves some theorems about them. For example, $1$-categories are just simplicial sets which are isomorphic to a nerve of a category (prop. 2.3.4.5). Moreover, every quasi-category $\mathcal{C}$ has an $n$-truncation $h_n\mathcal{C}$ which is an $n$-category with a map $\mathcal{C} \to h_n\mathcal{C}$ which is universal from $\mathcal{C}$ to an $n$-category (see prop. 2.3.4.12 for a precise statement).

Qusetion:Is there a model structure on the category of simplicial sets with monomorphisms as cofibrations, $n$-categories as fibrant objects and such that two quasi-categories are equivalent in this model structure if and only if their $n$-truncations are equivalent in the Joyal model structure?

I ask this mainly out of pure curiosity and haven't put too much effort in thinking about it myself, in the hope that someone already knows the answer. Hence, perhaps there are simple reasons that this can't happen, in this case (and any other case), I'm open to considering some variations.

**Remark**: In fact, as I learned from this question, the cofibrations and fibrant objects determine the model strucute, so the question can be broken in two. First, whether there is a model structure with the specified cofibrations and fibrant objects, and if so, how can we interpret the weak equivalences.