The enriched-category model conjecture for cartesian model categories

Let $C$ be a small category, let $

$$M=s\operatorname{PSh}(C)^{\mathrm{inj}}_{\mathscr{S}},$$

and suppose that $M$ is a cartesian model category. For an object $X$ of $M\operatorname{-Cat}$, let $hX$ be the ordinary category with the same objects as $X$ and morphism sets $hX(a,b)=hM(1,X(a,b))$. Call a morphism $f:X\to Y$ a weak equivalence if each map $X(a,b)\to Y(fa,fb)$ is a weak equivalence in $M$ and the induced functor $hX\to hY$ is an equivalence of categories. Enriched-category model conjecture. There is a model category structure on $M\operatorname{-Cat}$ with these weak equivalences, together with a Quillen equivalence

$$M\operatorname{-Cat}\simeq M\operatorname{-}\Theta\operatorname{Sp}.$$

This would identify the proposed $$\Theta$-space models with categories enriched over a cartesian model category, generalizing the relationship between complete Segal spaces and enriched homotopy theories. The conjecture is presented as a proposed model structure and Quillen equivalence; no resolution is given here.