Let $\pi : \mathcal T \to \mathcal A$ be a homological functor from a stable / triangulated category to an abelian category, and let $\mathcal C \subseteq \mathcal A$ be a weak Serre subcategory. Let $\pi^{-1}(\mathcal C) \subseteq \mathcal T$ be the full subcategory of objects $T$ such that $\pi_n(T) \in \mathcal C$ for all $n \in \mathbb Z$. Then $\pi^{-1}(\mathcal C)$ is a thick subcategory of $\mathcal T$ ("by construction" -- this is kind of the whole point of weak Serre classes!)

Question: Suppose that $\mathcal P \subseteq \mathcal T$ is a thick subcategory of a stable / triangulated category $\mathcal T$. Under what conditions does there exist an abelian category $\mathcal A$, a homological functor $\pi : \mathcal T \to \mathcal A$, and a weak Serre class $\mathcal C \subseteq \mathcal A$ such that $\mathcal P = \pi^{-1}(\mathcal C)$?

I'm also interested in the version of this question where $\mathcal T$ is compactly-generated, $\mathcal A$ is required to be Grothendieck, and $\mathcal \pi$ is required to commute with filtered colimits.