BHHT levelwise orbifold Euler characteristic 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 $\chi^{{\,\operatorname{orb}},[\sigma]}$ for the contribution to the orbifold Euler characteristic from elements whose permutation component lies in $[\sigma]$. BHHT levelwise orbifold Euler characteristic conjecture. For every conjugacy class $[\sigma]$ in $S$,

$$\chi^{{\,\operatorname{orb}},[\sigma]}(V_f,G\rtimes S)=(-1)^n\chi^{{\,\operatorname{orb}},[\sigma]}(V_{\widetilde{f}},\widetilde{G}\rtimes S).$$

This conjecture refines the known mirror-symmetry relation for the total orbifold Euler characteristic by asserting symmetry separately on each level, namely each conjugacy class of permutations. The paper reports indications but does not establish the general statement.