Kawamata's LG-model conjecture on FCY admissible subcategories

Let $(Y_1,w_1)$ and $(Y_2,w_2)$ be gauged Landau–Ginzburg models with $Y_i$ smooth, a group $G$ acting on each $Y_i$, and $w_i$ a section of $\operatorname{\mathcal O}(\chi)$ for a character $\chi:G\to\mathbb G_m$. Say that $(Y_1,w_1)$ $K$-dominates $(Y_2,w_2)$ if there exist a smooth $G$-variety $Z$ and proper equivariant birational morphisms $f_i:Z\to Y_i$ such that $f_1^*w_1=f_2^*w_2$ and $f_1^*K_{Y_1}-f_2^*K_{Y_2}\geq0$. Then, if $(Y_1,w_1)$ $K$-dominates $(Y_2,w_2)$ and $[Y_2/\operatorname{ker}\chi_2]$ has torsion canonical bundle, the category $\operatorname{D}^{\operatorname{abs}}[Y_2,G,w_2]$ is a fractional Calabi–Yau admissible subcategory of $\operatorname{D}^{\operatorname{abs}}[Y_1,G,w_1]$. This is presented as a specialization of Kawamata's LG-model conjecture and gives a proposed mechanism for producing fractional Calabi–Yau admissible subcategories; its resolution status is not specified in the supplied text.