Fukaya-category Hochschild–Floer isomorphism conjecture

Let $X$ be a compact symplectic manifold of dimension $2d$, and let $\operatorname{Fuk}(X)$ be its Fukaya category. The open-closed Gromov–Witten theory is expected to give a map from the closed TCFT associated to $\operatorname{Fuk}(X)$ to the Gromov–Witten TCFT of $X$, inducing a map

$$HH_\ast(\operatorname{Fuk}(X))\longrightarrow H_{\ast+d}(X).$$

Fukaya-category Hochschild–Floer isomorphism conjecture. In good circumstances, this map is an isomorphism. This would identify the Hochschild homology of the Fukaya category with the shifted homology of $X$; the source states the needed conditions only as “good circumstances.”