Let $\mathcal C$ be a presentable $\infty$-category. Then the stabilization $Stab(\mathcal C)$ of $\mathcal C$ is the universal presentable stable category on $\mathcal C$.

Conversely, if $\mathcal T$ is a presentable stable $\infty$-category, then we can ask which presentable $\infty$-categories $\mathcal C$ have $Stab(\mathcal C) \simeq \mathcal T$. There's always at least one such $\mathcal C$, namely $\mathcal T$ itself. In particular, I would like to know an answer to the following:

Question 1: Let $\mathcal T$ be a presentable stable $\infty$-category. Under what conditions does there exist an $\infty$-topos $\mathcal C$ such that $Stab(\mathcal C) \simeq \mathcal T$?

For a closely related question, let $StPr^L$ denote the $\infty$-category of presentable stable $\infty$-categories and left adjoint functors. Let $Logoi$ denote the $\infty$-category of $\infty$-topoi, with geometric morphisms pointing in the direction of their inverse images.

Question 2: Does the functor $Stab : Logoi \to StPr^L$ have a left or right adjoint?

If the answer to Question 2 is affirmative, then one might approach Question 1 by asking for criteria ensuring that the unit / counit of the adjunction is an equivalence. Alternatively, one might wonder

Question 3: Note that the functor $Stab : Pr^L \to StPr^L$ has both a left adjoint $L$ and a right adjoint $R$. For which presentable stable $\infty$-categories $\mathcal T$ is $L\mathcal T$ or $R \mathcal T$ an $\infty$-topos?