Let $\mathcal{M}$ be a locally presentable category equipped with two left proper and left determined combinatorial model structures $\mathcal{M}_1$ and $\mathcal{M}_2$. There exist two sets $S_1$ and $S_2$ such that the identity functor of $\mathcal{M}$ induces a Quillen equivalence $\mathbf{L}_{\mathcal{S_1}} \mathcal{M}_1 \simeq \mathbf{L}_{\mathcal{S_2}} \mathcal{M}_2$ between the left Bousfield localizations.

I am looking for examples of such a situation, keywords related to it, for what people managed to prove, the context etc...