Gross–Pavanelli mirror conjecture for the Heisenberg quotient Calabi–Yau threefold

Let $X$ be the Calabi–Yau threefold with fibers that are $(1,8)$-polarized abelian surfaces over $\mathbb{P}^{1}$, and let the free $\mathbb{Z}_{8}$ and $\mathbb{Z}_{8}\times\mathbb{Z}_{8}$ actions define

$$\check{X}:=X/\mathbb{Z}_{8},\qquad Y:=X/(\mathbb{Z}_{8}\times\mathbb{Z}_{8}).$$

Gross–Pavanelli's mirror conjecture. The mirror of $X$ is $\check{X}$, and the mirror of $\check{X}$ is $Y$.

The conjecture proposes a concrete sequence of mirror partners among Calabi–Yau threefolds obtained from the Heisenberg-group quotients. The supplied status evidence says that the conjecture was verified affirmatively by constructing the relevant families and studying large-complex-structure-limit boundary points.