Thomas–Yau Bridgeland stability conjecture for dHYM and special Lagrangians

Let $X$ be a Calabi–Yau manifold, let $L\rightarrow X$ be a holomorphic line bundle, and let $L\hookrightarrow X$ be a Lagrangian submanifold. The objects $L$ are viewed respectively in $D^b\operatorname{Coh}(X)$ and $D^b\operatorname{Fuk}(\check{X})$. Thomas–Yau Bridgeland stability conjecture. The holomorphic line bundle $L$ admits a metric solving the deformed Hermitian–Yang–Mills equation if and only if $L$ is stable in the sense of Bridgeland as an object in $D^b\operatorname{Coh}(X)$; likewise, the Lagrangian $L$ can be deformed by Hamiltonian deformations to a special Lagrangian if and only if $L$ is stable in the sense of Bridgeland as an object in $D^b\operatorname{Fuk}(\check{X})$. This conjecture proposes that Bridgeland stability algebraically characterizes solutions of the deformed Hermitian–Yang–Mills equation and special Lagrangians. The source presents it as a fundamental conjecture; no resolution is supplied here.