Hosono–Lee–Lian–Yau Hodge number duality conjecture for singular Calabi–Yau mirror pairs

Let $(\widetilde{T}_\Delta,\widetilde{T}_{\check{\Delta}})$ be the pair of $d$-dimensional singular Calabi–Yau varieties obtained as branched double covers of the toric varieties associated to dual nef-partition data. For $p,q\in\mathbb{Z}$, write $h^{p,q}$ for their usual Hodge numbers. Hosono–Lee–Lian–Yau conjecture. The pair satisfies the Hodge number duality

$$h^{p,q}(\widetilde{T}_\Delta)=h^{d-p,q}(\widetilde{T}_{\check{\Delta}}).$$

This duality is intended to make $(\widetilde{T}_\Delta,\widetilde{T}_{\check{\Delta}})$ a mirror pair. The case $d=3$ was proven in the work introducing these varieties, while the general case remains open.