Batyrev--Borisov homological mirror symmetry conjecture

Let $(Z,\check{Z})$ be a Batyrev--Borisov mirror pair, with $Z$ and $\check{Z}$ smooth subvarieties of Fano toric varieties $X_{\Sigma}$ and $X_{\check{\Sigma}}$ associated to dual nef partitions $\Delta=\Delta_1+\dots+\Delta_r$ and $\nabla=\nabla_1+\dots+\nabla_r$, and centred refined triangulating functions $h$ and $\check{h}$. Write $\mF$ for the Fukaya category and $D^b_{dg}$ for the dg-enhanced bounded derived category. Batyrev--Borisov HMS conjecture. There exist quasi-equivalences

$$\mF(Z)\cong D^b_{dg}(\check{Z}),\qquad \mF(\check{Z})\cong D^b_{dg}(Z).$$

This is presented as the ultimate goal of the paper and is intentionally somewhat imprecise. The preceding Hodge-number equalities provide evidence for mirror symmetry, but the stated categorical equivalences are not established in the source.