The torus-generation conjecture for Fukaya categories of homogeneous varieties

Let $G$ be a simple simply connected complex algebraic group, let $P\subset G$ be a parabolic subgroup, and let $V^\vee\subset G^\vee/P^\vee$ be the top projected open Richardson stratum in the Langlands-dual homogeneous variety. Let $\mathbf{D}\operatorname{Fuk}_\lambda(G/P)$ denote the Fukaya category for $\lambda\in\mathbb{C}$. The torus-generation conjecture. If the deep locus of $V^\vee$ is empty, then for every $\lambda\in\mathbb{C}$ the category

$$\mathbf{D}\operatorname{Fuk}_\lambda(G/P)$$

is generated by objects supported on Lagrangian tori. The conjecture is motivated by identifying cluster charts of $V^\vee$ with spaces of local systems on Lagrangian tori and their superpotentials with disk potentials. The construction and generation argument remain open; the paper explicitly declines to make the claim when the deep locus is nonempty.