Consider the model structure on simplicial sets where the cofibrations are given by monomorphisms and the weak equivalences are given by $n$-equivalences (that is, maps $f \colon X \to Y$ that induce a bijection $f_{*} \colon \pi_{0}X \to \pi_{0}Y$ and isomorphisms $f_{*} \colon \pi_{k}(X,x) \to \pi_{k}(Y,fx)$ for all $0 < k \leq n$ and all $x \in X$).

This can be constructed variously as the left Bousfield localization of the standard Kan-Quillen model structure on simplicial sets, or as a Cisinski model structure where the set of generating anodyne extensions is given by:
$$J_{n} = \left\{\Lambda^{k}_{i} \hookrightarrow \Delta^{k} \mid k > 0, i = 0, \ldots, k \right\} \cup \left\{\partial\Delta^{k} \hookrightarrow \Delta^{k} \mid k \geq n+2 \right\}$$

We know, for instance, that the fibrant objects are $n$-truncated Kan complexes. We also know that a map with fibrant codomain is a fibration if and only if it has the right lifting property against every element of $J_{n}$.

My question is this: do we know whether or not an arbitrary map with right lifting property against every element of $J_{n}$ is a fibration in this model structure? If not, is there a simple counterexample that proves it to be false?