HMS of log Calabi–Yau varieties over the Novikov ring

Let $(X,D)$ be a log Calabi–Yau manifold, with $D$ an anticanonical divisor of $X$, and let $W_X\colon \check X\to\mathbb C$ be a mirror of $(X,D)$. Let $\widehat{\Lambda}_{\geq}$ be the Novikov ring, and let $\operatorname{Fuk}(W_X,\widehat{\Lambda}_{\geq})$ be the derived Fukaya category over $\widehat{\Lambda}_{\geq}$ generated by Lagrangians compactly Hamiltonian isotopic to Lefschetz thimbles of $W_X$. Let $\sqrt[\infty]{X}$ and $\sqrt[\infty]{D}$ be the infinite root stack of $(X,D)$, and let $\pi^B$ be the displayed functor to $\operatorname{aQCoh}(\sqrt[\infty]{X_\Sigma},\sqrt[\infty]{D},\widehat\Lambda)$. HMS of log CY over the Novikov ring. There exists a $\widehat{\Lambda}_{\geq}$-linear embedding

$$\operatorname{Fuk}(W_X,\widehat{\Lambda}_{\geq})\hookrightarrow (\pi^B)^{-1}(\operatorname{QCoh}(X,\widehat{\Lambda})).$$

This is a proposed homological mirror symmetry statement for log Calabi–Yau manifolds, relating the Fukaya category generated by Lefschetz thimbles to a subcategory of quasicoherent sheaves on the corresponding infinite-root-stack side. Its status is not resolved in the supplied text.