Homological Berglund–Hübsch–Henningson mirror symmetry for invertible polynomials

Let $\mathbf{w}$ be an invertible polynomial with admissible symmetry group $\Gamma\subseteq\Gamma_{\mathbf{w}}$, and let $\widecheck{\Gamma}=\operatorname{Hom}(\Gamma_{\mathbf{w}}/\Gamma,\mathbb{C}^*)$ be the corresponding dual group. Write $\widecheck{\mathbf{w}}$ for the mirror polynomial, and let $\mathrm{mf}(\mathbb{A}^n,\Gamma,\mathbf{w})$ and $\mathcal{FS}(\widecheck{\mathbf{w}},\widecheck{\Gamma})$ denote the associated pre-triangulated $A_\infty$-categories over $\mathbb{C}$. Homological Berglund–Hübsch–Henningson mirror symmetry. There is a quasi-equivalence

$$\mathrm{mf}(\mathbb{A}^n,\Gamma,\mathbf{w})\simeq \mathcal{FS}(\widecheck{\mathbf{w}},\widecheck{\Gamma}).$$

This predicts an equivalence between the equivariant matrix-factorisation category of an invertible polynomial and the Fukaya–Seidel category of its Berglund–Hübsch–Henningson mirror. The statement is presented as a prediction in the source, and its general validity remains open.