There are a few fibrant replacement functors for the Quillen model structure on simplicial sets that preserve cartesian products, namely $\operatorname{Ex}^\infty$ and $\operatorname{Sing}(|\cdot|)$. Does there exist a product-preserving functorial fibrant replacement for the Joyal model structure?

There are a few fibrant replacement functors for the Quillen model structure on simplicial sets that preserve finite cartesian products, namely $\operatorname{Ex}^\infty$ and $\operatorname{Sing}(|\cdot|)$. Does there exist a product-preserving functorial fibrant replacement for the Joyal model structure?