Higher-rank Landau–Ginzburg compactification conjecture

Let $X$ be a quasi-Fano variety with a smooth anticanonical divisor $D$, and let $D_1+D_2$ be a simple normal crossings anticanonical divisor of $X$. Let $(X^\beth,h)$ denote the rank $2$ Landau–Ginzburg mirror of $(X,D_1+D_2)$, and let $(\mathbb P_{D_1}(N_{D_1}\oplus\mathcal O),D_1+\tilde D_1)$ be the associated normal-cone component. Landau–Ginzburg compactification conjecture. The ordinary Landau–Ginzburg model of $(X,D)$ is a partial compactification of the rank $2$ Landau–Ginzburg model of $(X,D_1+D_2)$, obtained by gluing the rank $2$ Landau–Ginzburg mirror of $(\mathbb P_{D_1}(N_{D_1}\oplus\mathcal O),D_1+\tilde D_1)$ to the rank $2$ Landau–Ginzburg mirror of $(X,D_1+D_2)$. This describes mirror symmetry for degeneration to the normal cone, where the non-proper rank $2$ model is compactified to the ordinary model; the paper uses this conjecture but gives no resolution.