I came along the following question while trying to understand and apply some ideas of Dugger's article *Universal Homotopy Theories*.

Suppose, we are given a nice model category $\mathcal{C}$, say left proper and cellular or combinatorial, so we have a good theory of localization. I am primarly thinking here of the category of presheaves of simplicial sets on some site with the projective model structure, where weak equivalences and fibrations are defined "pointwise".

Suppose furthermore, $S$ is a class of morphisms in $\mathcal{C}$ we can (left Bousfield) localize at [e.g. the class required for descent for hypercovers] and $T$ is an arbitrary set of morphisms in $\mathcal{C}$.

Now, let's consider a fibrant object $X$ in $\mathcal{C}[S^{-1}]$, i.e. some object which is $S$-local (and $\mathcal{C}$-fibrant), and take it's fibrant replacement $X^f$ in $\mathcal{C}[S^{-1}][T^{-1}]$.

Is it now reasonably to expect under some circumstances or even generally true that the map $X \to X^f$ is a weak equivalence in $\mathcal{C}[T^{-1}]$?