Suppose we have a compactly generated triangulated category $\mathcal{T}$ such that the subcategory of compact objects $\mathcal{T}^c$ is essentially small. Let us take $\mathcal{A}, \mathcal{B}$ two thick subcategories of compact objects. I was wondering if the equality $\text{Loc}(\mathcal{A})\cap \text{Loc}(\mathcal{B})=\text{Loc}(\mathcal{A}\cap\mathcal{B})$ holds. Here $\text{Loc}(S)$ denotes the localizing subcategory generated by $S$ a class of objects of $\mathcal{T}$.
Since $\mathcal{T}$ is compactly generated Thomason's Localization Theorem states that if $S$ is a set of compact objects then $\text{Loc}(S)\cap \mathcal{T}^c=\text{Thick}(S)$, where the last term is the thick subcategory generated by $S$. Using this we easily deduce that both $\text{Loc}(\mathcal{A})\cap \text{Loc}(\mathcal{B})$ and $\text{Loc}(\mathcal{A}\cap\mathcal{B})$ have the same class of compact objects, namely $\mathcal{A}\cap \mathcal{B}$.
If also $\text{Loc}(\mathcal{A})\cap \text{Loc}(\mathcal{B})$ were generated by its compact objects, then the equality would follow. But I do not know if this is the case. I do not even understand if it is a reasonable expectation or counterexamples can be found easily. Any pointer or observation is welcome.