Double mirror conjecture for affine log Calabi–Yau varieties

Let $U$ be an affine log Calabi–Yau variety and let $V$ be a general fibre of its mirror family; assume the mirror construction extends to the canonical-singularity setting considered in the source. Let $\operatorname{Sk}(V)$ denote the skeleton of $V$ and let $\mathcal O(U)$ be the ring of regular functions on $U$. Double mirror conjecture. $U$ is a fibre of the mirror family for $V$. In particular, $\operatorname{Sk}(V)$ parameterizes a basis of $\mathcal O(U)$, canonical up to individual scaling.

The conjecture is described as the ultimate goal of the project: proving that affine log Calabi–Yau varieties with maximal boundary have canonical bases of regular functions. The corresponding statement is known in dimension two, while the general case remains open.