Batyrev–Borisov-type mirror conjecture for log Calabi–Yau varieties

Let $U$ and $V=U^\star$ be mirror log Calabi–Yau varieties with theta-function bases, and let $P\subset N_U$ and $Q=P^\star\subset M_U$ be dual reflexive polygons. The polytope $P$ determines a Landau–Ginzburg model

$$w_P=\sum_{p\in P}a_p\varphi_p,$$

with coefficients $a_p\in\mathbb{Z}_{\geq 0}$, while $Q$ determines a grading on $\mathbb{C}[U]$ and hence a compactification $(X_Q,D_Q)$ with $X_Q=\operatorname{Proj}R_Q$. Batyrev–Borisov-type mirror conjecture. If $(X_Q,D_Q)$ admits a $\mathbb{Q}$-Gorenstein deformation to a pair $(X,D)$ with $X$ a $\mathbb{Q}$-factorial Fano variety, then there exists a choice of positive integral coefficients on the lattice points of $P$ such that $(V,w_P)$ is mirror to $(X,D)$. This is proposed as a generalisation of Batyrev–Borisov mirror symmetry from toric Fano varieties to the log Calabi–Yau setting; the supplied text gives no resolution status.