Takahashi's homological mirror symmetry conjecture for invertible polynomials

Let $f\in S^n$ be an invertible polynomial, and let $\widetilde{f}$ denote its Berglund–Hübsch transpose. Let $L_f$ be the maximal grading of $f$, and let $\operatorname{HMF}^{L_f}_{S^n}(f)$ be the homotopy category of maximally graded matrix factorizations. Write $\operatorname{D^b}\operatorname{Fuk}^{\to}(\widetilde{f})$ for the derived category of the directed Fukaya category of $\widetilde{f}$. Takahashi's homological mirror symmetry conjecture. There is a finite acyclic quiver $Q$ with admissible relations $I$ such that

$$\operatorname{HMF}^{L_f}_{S^n}(f)\simeq \operatorname{D^b}(\operatorname{mod}\,\mathbb{C}Q/I)\simeq \operatorname{D^b}\operatorname{Fuk}^{\to}(\widetilde{f}).$$

This is a version of homological mirror symmetry for isolated hypersurface singularities. It is known for ADE polynomials, Brieskorn–Pham singularities, certain Thom–Sebastiani sums, and the cases with $n=2$, but remains open in general.