The category $\mathrm{Set}_{\Delta}^{+}$ of marked simplicial sets has a model structure (the (co)cartesian model structure) constructed by Lurie in HTT.3.1.3. I would like to know if this model structure admits a fibrant replacement functor $R$ that is lax monoidal with respect to the cartesian product of marked simplicial sets, that is, for which there are natural maps (necessarily equivalences) $$ R(X) \times R(Y) \to R(X \times Y), $$ which commute with the fibrant replacement maps coming from $X \times Y$ and which are associative and unital in the usual sense.