I'm still unclear on what the motivation for looking at such a model structure might be, so I'm going to go ahead and guess that what you're ultimately interested in, Fosco, is an alternate construction of the usual model structure on $\mathrm{Set}_\Delta^+$ constructed in HTT 3.1.3.7 (where we take $S$ to be a point). Apologies if I'm guessing wrong!

**Verity's approach:**

Verity almost does this. To see this, recall that a *stratified simplicial set* in Verity's sense is a simplicial set equipped with certain "thin" simplicies which can have arbitrary dimension $\geq 1$ (and it's required that every degenerate simplex is thin). Verity constructs a model structure on the category $\mathrm{Strat}$ of stratified simplicial sets whose fibrant objects are what Verity calls *weak complicial sets*. A marked simplicial set can be considered as a stratified simplicial set by taking a 1-simplex to be thin iff it is marked, and taking all simplices of dimension $\geq 2$ to be thin. Then Verity's model structure on $\mathrm{Strat}$ restricts to a model structure on $\mathrm{Set}_\Delta^+$.

The only caveat is that Verity's model structure needs to be further localized, by a "Rezk completeness condition". That is, in his model structure, every fibrant marked simplicial set will have the property that every marked simplex is an equivalence, but will not necessarily have the converse property that every equivalence is marked. This needs to be enforced by a Bousfield localization. This is not hard to do, but I'm not sure it's been done in the literature.

**The Cisinski/Olschok approach:**

It's notable that Verity's construction of his model structure is very much analogous to Joyal's construction of the Joyal model structure, which in turn is essentially an application of Cisinski's theory of model structures on Grothendieck topoi. The only reason that Cisinski's theory can't be directly applied to marked simplicial sets is that marked simplicial sets don't actually form a topos! But that's okay -- Olschok has extended Cisinski's theory to general locally presentable categories. Olshok's theory is a very general method for constructing model structures from a set of generating cofibrations, a set of *elementary anodyne extensions*, and a *functorial cylinder object*.

In order to apply Olschok's theory, we must simply identify a class of elementary anodyne extensions and a functorial cylinder object (it's easy to write down a set of generating cofibrations such that the cofibrations are exactly the monomorphisms). The elementary anodyne extensions from Verity's paper (supplemented by a morphism corresponding to Rezk completeness), and the functorial cylinder given by $X \mapsto X \times \Delta[1]_t$ (where $\Delta[1]_t$ is the marked 1-simplex) yield the usual model structure. Olschok's theory allows us to identify the fibrant objects and fibrations between fibrant objects in this model structure, via lifting against the elementary anodyne extensions, if we additionally verify that the anodyne extensions are closed under certain pushouts involving the functorial cylinder; This is a straightforward combinatorial exercise.