Gross–Hacking–Keel theta-function conjecture for positive log Calabi–Yau varieties

Let $U$ be a positive log Calabi–Yau variety with maximal boundary, and let $N_U(\mathbb{Z})$ denote its integral tropical points. A mirror positive log Calabi–Yau variety is a positive log Calabi–Yau variety $V$ with maximal boundary, together with a coordinate ring whose additive basis is indexed by $N_U(\mathbb{Z})$. Gross–Hacking–Keel conjecture. There exists such a mirror variety $V$ whose coordinate ring $\mathbb{C}[V]$ has an additive basis

$$\mathcal B_V=\{\vartheta_n\in\mathbb{C}[V]:n\in N_U(\mathbb{Z})\},$$

where the basis elements are theta functions, canonically determined up to multiplication by scalars, and whose multiplication structure constants are obtained as certain counts of rational curves in $U$. This conjecture seeks to generalise the duality between the character and cocharacter lattices of mirror algebraic tori to pairs of mirror log Calabi–Yau varieties. The claimed theta basis and curve-counting multiplication are central to the Gross–Hacking–Keel program for mirror symmetry; the supplied text gives no resolution status.