Bridgeland-type stability conjecture for symplectic autoequivalences of cubic K3 categories

Let $X\subset\mathbb P^5$ be a smooth cubic fourfold, let $\mathcal A_X$ be its K3 category, and let $\operatorname{Aut}_{\rm s}(\mathcal A_X)$ denote the group of symplectic autoequivalences. Let $P_0$ and ${\rm O}$ be the space and group appearing in the stability-condition formulation referenced by the source, and let $\pi_1^{\rm st}[P_0/{\rm O}]$ denote the corresponding stability fundamental group. Bridgeland-type conjecture. There exists an isomorphism

$$\operatorname{Aut}_{\rm s}(\mathcal A_X)\simeq\pi_1^{\rm st}[P_0/{\rm O}].$$

The statement is presented as a reformulation inspired by Bridgeland's conjecture for K3 surfaces. Because the supplied context does not define $P_0$, ${\rm O}$, or the superscript ${\rm st}$, and gives no resolution status, the conjecture should be checked against the full paper.