Suppose that $\mathscr{P}$ is a (locally) presentable $\left(\infty,1\right)$ category (which we can assume WLOG is infinity presheaves on some small $\left(\infty,1\right)$ category) , and $R$ and $S$ are classes of morphisms of $\mathscr{P}$ with respect to which we want to localize. Let us not assume that either $R$ or $S$ are a priori saturated classes. Let $\mathscr{P}_S$ and $\mathscr{P}_R$ denote the full subcategory on $S$-local objects and $R$-local object respectively. Their respective inclusions into $\mathscr{P}$ have left adjoints $L_R$ and $L_S$.
Question: On what conditions on $S$ and $R$ (besides $S \subset R$) will $L_R$ map $\mathscr{P}_S$ to itself?
(An answer even for $1$-categories would be very helpful).
Remark: I'm most interested in the case where $\mathscr{P}$ is $\infty$-presheaves on a Grothendieck site, $S$ consists of the associated covering sieves, and $R$ is in the essential image of the Yoneda embedding.