Folklore stability conjecture for supersymmetric branes

Let $X$ be a Calabi–Yau manifold with mirror $\check{X}$, let $L$ be a Lagrangian in $\check{X}$, and let $E$ be a holomorphic vector bundle on $X$. Write $D^{b}{\rm Fuk}(\check{X})$ for the derived Fukaya category and $D^{b}\operatorname{Coh}(X)$ for the bounded derived category of coherent sheaves. A Lagrangian is special Lagrangian if it can be represented by a special Lagrangian, and a bundle admits a deformed Hermitian–Yang–Mills metric if it admits a metric solving the deformed Hermitian–Yang–Mills equation. Folklore conjecture. There is a Bridgeland stability condition on $D^{b}{\rm Fuk}(\check{X})$ (respectively, $D^{b}\operatorname{Coh}(X)$) such that the isomorphism class of $L$ (respectively, $E$) is stable if and only if it contains a special Lagrangian (respectively, $E$ admits a metric solving the deformed Hermitian–Yang–Mills equation). This is a proposed categorical formulation of the correspondence between stability and supersymmetric $A/B$-branes; the source gives no resolution of the conjecture.