Is a left Bousfield localization of simplicial presheaves a locally cartesian closed model category?

Question

Let $\mathcal{C}$ be a small category and let $\mathcal{M} = \operatorname{sPre}(C)$ be the model category of simplicial presheaves on $\mathcal{C}$ with the injective model structure. Let $S$ be a set of morphisms in $\mathcal{M}$ that is stable under pullback and let $L_S\mathcal{M}$ be the left Bousfield localization of $\mathcal{M}$ with respect to $S$.

By this nlab page, $\mathcal{M}$ is a locally cartesian closed model category, and any right proper left Bousfield localization of $\mathcal{M}$ is as well. Is this true for any other left Bousfield localization $L_S\mathcal{M}$? If not, is it true when $S$ is (the saturation of) the set of covering sieves $R \hookrightarrow j(X)$ for some Grothendieck topology $t$ on $\mathcal{C}$, where $j(X)$ is the simplicial Yoneda embedding applied to $X \in \mathcal{C}$ (note that the local objects of $L_S\mathcal{M}$ are simplicial presheaves satisfying $t$-descent but not necessarily $t$-hyperdescent)? This is really the case I care about. If not, is there a counterexample?

It should be noted that $L_S\mathcal{M}$ is at least Quillen equivalent to a locally cartesian closed model category because the $\infty$-category it presents is locally cartesian closed by a result of Hoyois.

Answer 1

Section 2 of this paper of Rezk addresses exactly the question of when the localization by S yields a Cartesian model category. For that the relevant property is that that if you take the product of a map in S and a representable object, the result must be in the saturation of S.

There are examples of S which fail this condition where the resulting localization is not a Cartesian model category. (An example is the n-fold Segal space model category constructed by Barwick - it is a simplicial model category but not Cartesian).

For local Cartesian-ness you are essentially talking about Cartesian-ness in the over categories $\mathcal{M}_{/X}$. There the same argument works where stability under products with representables is replaced with stability under fiber products of representables over X. So if your set of morphisms S is stable under all pullbacks, as mentioned in the OP, that is sufficient to give the local cartesian model category structure on $L_S\mathcal{M}$.