Berglund–Hübsch–Krawitz graded mirror symmetry conjecture for Milnor fibers

Let $V$ be the Milnor fiber under consideration, let $W^\vee$ be its Berglund–Hübsch mirror polynomial, and let $\Gamma^\vee$ be the maximal symmetry group of $W^\vee$,

$$\Gamma^\vee:=\{(t_1,\ldots,t_n,t_{n+1})\in (\mathbb{C}^\times)^{n+1}\mid W^\vee(t_1z_1,\ldots,t_nz_n)=t_{n+1}W^\vee(z_1,\ldots,z_n)\}.$$

Write $\mathbb{C}_{x_0}\times\mathbb{C}^n$ for affine space with coordinates $x_0,x_1,\ldots,x_n$, and let $\operatorname{MF}^{\Gamma^\vee}$ denote the dg category of $\Gamma^\vee$-equivariant matrix factorizations. Berglund–Hübsch–Krawitz graded mirror symmetry conjecture. There is an equivalence of dg categories

$$\mathcal{W}(V)\cong\operatorname{MF}^{\Gamma^\vee}(\mathbb{C}_{x_0}\times\mathbb{C}^n,x_0x_1\cdots x_n+W^\vee).$$

This is the conjectural $\mathbb{Z}$-graded version of the mirror description of the wrapped Fukaya category of the Milnor fiber. The source presents it as the main conjecture of the cited work; the present paper proves a related statement conditionally on this expected property, while a complete account is outside its scope.