Family Floer cohomology and mirror symmetry conjecture

Let $X$ be the symplectic manifold under consideration. For each Lagrangian $L_i\subset X$, let $\mathcal L_i$ be the corresponding object in the category of $\alpha_X$-twisted sheaves of perfect complexes on the mirror $Y$, and let $HF^*(L_1,L_2)$ denote their Floer cohomology. Family Floer cohomology conjecture. There is an isomorphism

$$HF^{*}(L_1,L_2) \cong H^*(\operatorname{Hom}_{*}(\mathcal L_1,\mathcal L_2)).$$

This conjecture asserts that the differential graded objects produced by family Floer theory recover Floer cohomology through morphisms of twisted sheaves of perfect complexes, explaining why the complexes themselves, rather than only their cohomology sheaves, are the appropriate mirror objects. The surrounding discussion presents this as a conjectural direction following the construction of the corresponding twisted sheaves.