Homological mirror symmetry for symmetric squares of punctured spheres

Let $V_{n,k}$ be the symmetric-square-type Liouville space considered in the paper, equipped with its natural integer grading $\eta_0$, and let $\mathcal{Z}_{n,k}$ be the special fiber of the smooth toric variety $\mathcal{Y}_{n,k}$ over $V_1=\cdots=V_k=0$. Let $\mathcal{X}_{n,k}=\mathcal{Y}_{n,k}\times\operatorname{Spec}\mathbf{k}[U_1,\ldots,U_k]$ with Landau–Ginzburg potential $\mathbf{w}_{n,k}=U_1V_1+\cdots+U_kV_k$. Write $\mathcal{B}_{n,k}$ for the category of $\mathbb{L}_B$-graded equivariant matrix factorizations of $\mathbf{w}_{n,k}$ on $\mathcal{X}_{n,k}$, where $\mathbb{L}_B\simeq\mathbb{L}$ is the natural grading group. Homological mirror symmetry conjecture. There should exist quasi-equivalences

$$\mathcal{W}(V_{n,k},\eta_0)\simeq D^b\operatorname{Coh}(\mathcal{Z}_{n,k}),$$

and

$$\mathcal{W}(V_{n,k})\simeq\mathcal{B}_{n,k}.$$

Moreover, after considering deformations over $\mathbb{A}^k$, there should exist a quasi-equivalence

$$\mathcal{W}(T^*\mathbb{T}^n,D_{n,k},\eta_0)\simeq D^b\operatorname{Coh}(\mathcal{Y}_{n,k}).$$

This conjecture proposes the homological mirror for arbitrary $n$ and $k$, extending the previously established cases and relating wrapped Fukaya categories to coherent sheaves and graded matrix factorizations. Its resolution is not supplied in the source.