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.