Thomas-Yau's categorical semistability correspondence conjecture

Let $X$ be a Calabi-Yau manifold, let $D^bFuk(X)$ denote its derived Fukaya category, and let Joyce's conjectural Bridgeland stability be a stability condition on this category. An almost calibrated exact Lagrangian brane is an exact Lagrangian brane whose phase stays in an interval of length strictly less than $\c0$; a Thomas-Yau semistable brane is one satisfying the phase inequalities for all relevant distinguished triangles. Thomas-Yau-Joyce correspondence conjecture. An almost calibrated exact Lagrangian brane $L$ defines a semistable object in $D^bFuk(X)$ under Joyce's Bridgeland stability if and only if it is Thomas-Yau semistable. The claim identifies the geometric Floer-theoretic notion of semistability with categorical Bridgeland semistability; Joyce's stability condition and the required enlargement of the Fukaya category are themselves conjectural, so this correspondence is open.