The Keel–Yu mirror detropicalization conjecture

Let $U$ be an affine log-Calabi–Yau variety containing an open torus. Let $\mathrm{sk}(U)$ denote its Kontsevich–Soibelman tropical skeleton, and let $\mathcal{M}$ and $\mathcal{N}$ be polyptych lattices. A detropicalization of a polyptych lattice $\mathcal{M}$ is a pair $(\mathcal{A}_{\mathcal{M}},\mathfrak{v}:\mathcal{A}_{\mathcal{M}}\to\mathcal{S}_{\mathcal{M}})$ as in the paper; a strict dual pair is a pair of polyptych lattices satisfying the paper's strict-duality condition, and a chart-Gorenstein-Fano PL polytope is a PL polytope with the stated chart-Gorenstein-Fano property.

Keel–Yu mirror detropicalization conjecture. (1) In the presence of multiple open tori, the integral points of $\mathrm{sk}(U)$ can be equipped with the structure of a polyptych lattice $\mathcal{M}$. Moreover, the Keel–Yu mirror algebra can be realized as a detropicalization of $\mathcal{M}$. (2) In particular, if $\mathcal{M}$ and $\mathcal{N}$ are a strict dual pair of polyptych lattices, and $\mathcal{N}$ admits a detropicalization and a chart-Gorenstein-Fano PL polytope $\mathcal{P}$, then the coordinate ring of the Keel–Yu mirror to $\operatorname{Spec}(\mathcal{A}_{\mathcal{N}})$ is a detropicalization of $\mathcal{M}$.

This is presented as a suspected statement connecting the paper's polyptych-lattice detropicalizations with log-Calabi–Yau mirror constructions of Keel and Yu. The precise hypotheses under which the skeleton acquires the required polyptych structure and the mirror algebra realizes the detropicalization remain open.