The conjectural model structure on $A_{\infty}$-precat categories

Let ${\cal C}$ be an $A_{\infty}$-operad, and let $\widehat{{\cal C}}$-precats be the associated category of precategories. Assume the cofibrations are the maps defined in the preceding construction, and call a morphism $A\rightarrow B$ a weak equivalence when the induced map $Cat _{{\cal C}}(A)\rightarrow Cat _{{\cal C}}(B)$ is an equivalence of $\widehat{{\cal C}}$-categories.

Model-structure conjecture. For a wide range of $A_{\infty}$-operads ${\cal C}$, the category of $\widehat{{\cal C}}$-precats admits a closed model structure with these cofibrations and weak equivalences, and with fibrations the maps satisfying the lifting property with respect to trivial cofibrations.

Such a model structure would provide a homotopical framework for $\widehat{{\cal C}}$-categories and their precategorical presentations. The source does not specify the range of operads or provide evidence of resolution.