BHHT levelwise orbifold monodromy zeta-function mirror symmetry

Let $f$ be an invertible polynomial in $n$ variables, let $G\subset G_f$ be a group of diagonal symmetries, and let $S\subset S_n$ preserve $f$ and $G$. Let $(\widetilde{f},\widetilde{G}\rtimes S)$ be the BHHT-dual of $(f,G\rtimes S)$, and let $[\sigma]$ be the conjugacy class in $S$ of an element $\sigma\in S$. Assume that $S$ satisfies the parity condition (PC): for every subgroup $T\subset S$, $\dim(\mathbb{C}^n)^T\equiv n\pmod 2$. Write $\overline{\zeta}^{{\,\operatorname{orb}},[\sigma]}$ for the contribution to the orbifold monodromy zeta function from elements whose permutation component lies in $[\sigma]$. BHHT levelwise zeta-function conjecture. For every conjugacy class $[\sigma]$ in $S$,

$$\overline{\zeta}^{{\,\operatorname{orb}},[\sigma]}_{f,G\rtimes S}(t)=\left(\overline{\zeta}^{{\,\operatorname{orb}},[\sigma]}_{\widetilde{f},\widetilde{G}\rtimes S}(t)\right)^{(-1)^n}.$$

This is the levelwise refinement of the established zeta-function symmetry for abelian BHH-dual pairs. The paper gives evidence for the conjecture under PC but leaves the general assertion open.