Dolgachev–Nikulin duality conjecture for compactified Landau–Ginzburg models

Let $X$ be a smooth Fano threefold, and let $\mathsf{f}\colon Z\rightarrow\mathbb{P}^1$ be its log Calabi–Yau compactified toric Landau–Ginzburg model. Let $F$ be a general fibre of $\mathsf{f}$. The lattice $\operatorname{Pic}(X)^{\vee}$ is the Dolgachev–Nikulin dual lattice of the lattice polarization induced by $X$ on a smooth anticanonical K3 divisor, and $\operatorname{res}$ denotes restriction from $Z$ to $F$.

Dolgachev–Nikulin duality conjecture. There is an isomorphism of lattices

$$\operatorname{Pic}(X)^{\vee}\cong\operatorname{im}\left(H^2(Z,\mathbb{Z})\xrightarrow{\operatorname{res}}H^2(F,\mathbb{Z})\right).$$

This conjecture extends the known result for smooth toric Fano threefolds, where Ueda's argument applies using results on mirror symmetry for variations of integral Hodge structures. The analogous statement for general Fano threefolds is proposed because those results are not currently known in the required generality.