Berglund–Hübsch mirror symmetry conjecture for invertible polynomials

Let $A=(a_{ij})$ be an $(n+1)\times(n+1)$ integer matrix with nonzero determinant, and define

$${\bf w}(x_1,\ldots,x_{n+1})=\sum_{i=1}^{n+1}\prod_{j=1}^{n+1}x_j^{a_{ij}}.$$

Let $\check{{\bf w}}$ be the polynomial associated to $A^T$, let $\Gamma_{\bf w}$ be the finite extension of ${\mathbb G}_m$ defined by the weights preserving ${\bf w}$, and let $\mathcal F(\check{{\bf w}})$ denote the Fukaya–Seidel category of a Morsification of $\check{{\bf w}}$. Berglund–Hübsch mirror symmetry conjecture. There is a quasi-equivalence of idempotent complete $A_\infty$-categories

$$\mathcal F(\check{{\bf w}})\simeq \operatorname{mf}({\mathbb A}^{n+1},\Gamma_{\bf w},{\bf w}),$$

and there is a full collection of vanishing thimbles $\Delta_1,\ldots,\Delta_K$ such that, for $\mathcal A:=\operatorname{end}_{\mathcal F(\check{{\bf w}})}(\bigoplus_i\Delta_i)$, the $A_\infty$-algebra $\mathcal A$ is quasi-isomorphic to $A:=H(\mathcal A)$. Thus both categories are quasi-equivalent to $\operatorname{perf}(A)$. This is a homological mirror-symmetry statement for invertible polynomials; the source attributes versions to earlier work, but the general conjecture remains open.