The enriched model structure conjecture for marked simplicial spaces

Let $s\mathscr{S}\mathrm{et}$ be the category of simplicial sets, $s\mathscr{S}\mathrm{et}^+$ the category of marked simplicial sets, and $s\mathscr{S}^+$ the category of simplicial spaces. Let $i^+_1\colon s\mathscr{S}\mathrm{et}^+ \to s\mathscr{S}^+$ and $i_1\colon s\mathscr{S}\mathrm{et} \to s\mathscr{S}^+$ be the inclusions. Enriched model structure conjecture. There exists a model structure on $s\mathscr{S}^+$ satisfying the following conditions: it is Quillen equivalent to the Joyal model structure; $i^+_1$ is left Quillen from the Cartesian model structure; and $i_1$ is left Quillen from the Joyal model structure. This model structure would provide the foundation for a strictly quasi-categorically enriched version of the unstraightening construction, a generalization not supplied by the existing results.