Conservativity of the left Quillen strictification functor

For an integer $m$, let $\mathbf{sSet}^m$ denote the category of stratified simplicial sets and let $\infty\text{-}\mathbf{Cat}^m$ denote the category of marked strict $\infty$-categories. Let

$$|\_|: \mathbf{sSet}^m \to \infty\text{-}\mathbf{Cat}^m$$

be the left Quillen functor from the paper's construction. Strictification conservativity conjecture. The functor $|\_|$ reflects weak equivalences between cofibrant objects. This is presented as a concrete version of the conjecture that strictification is conservative on weak $(\infty,m)$-categories for all $m$; its resolution is not given here.