Kajiura–Saito–Takahashi conjecture for dual regular weight systems

Let $W$ be a regular weight system and $f$ be a quasi-homogeneous polynomial attached to $W$. Assume $W$ has a dual regular weight system $W^*=(a^*,b^*,c^*;h)$ in the sense of K. Saito, and let $f^*$ be a quasi-homogeneous polynomial attached to $W^*$. Write $D^b_{{{\mathbb Z}}}({\mathcal A}_{f^*})$ for the ${\mathbb Z}$-equivariant derived category of the matrix-factorization category associated with $f^*$. For objects $E_i$, let $\chi(E_i,E_j)=\dim_k\operatorname{Hom}_{D^b_{\mathbb Z}({\mathcal A}_{f^*})}(E_i,E_j)$, let $A=(a_{ij})$, and let $I_{K_0(D^b_{\mathbb Z}({\mathcal A}_{f^*}))}=A^{-1}+{}^tA^{-1}$. Kajiura–Saito–Takahashi conjecture. The following should hold: $D^b_{\mathbb Z}({\mathcal A}_{f^*})$ is generated by a strongly exceptional collection $\{E_1,\dots,E_\mu\}$; its Serre functor $S$ satisfies

$$S^h\simeq [3h-2a^*-2b^*-2c^*];$$

and the lattice $\bigl(K_0(D^b_{\mathbb Z}({\mathcal A}_{f^*})),I_{K_0(D^b_{\mathbb Z}({\mathcal A}_{f^*}))}\bigr)$ is isomorphic to $\bigl(H_2(X_1,{\mathbb Z}),-I_{H_2(X_1,{\mathbb Z})}\bigr)$. Here $[1]$ is the shift functor. This conjecture links equivariant derived categories of matrix factorizations for dual regular weight systems with exceptional collections, fractional Calabi–Yau behavior, and the lattice theory of singularities; the source does not establish its status.