Mirror-pair conjecture for gauge-fixed double covers from dual nef-partitions

Let $(\Delta,\{\Delta_i\}_{i=1}^{r})$ and $(\nabla,\{\nabla_i\}_{i=1}^{r})$ be dual nef-partitions, and let $X$ and $X^{\vee}$ be smooth maximal projective crepant partial desingularizations of the associated toric varieties. Choose sections $s_j\in \mathrm{H}^{0}(X,E_j)$ so that the toric boundary together with the divisors of the sections is a strict normal crossing divisor, and let $\mathcal{Y}\to V$ and $\mathcal{Y}^{\vee}\to U$ be the resulting gauge-fixed double-cover families; write $Y$ and $Y^{\vee}$ for their fibers. Mirror-pair conjecture. The pair $(Y,Y^{\vee})$ is a mirror pair. The claim is presented as a main result for gauge-fixed double covers branched along nef-partitions, but the supplied text gives no resolution status.