Homological mirror symmetry for compactified symmetric squares of punctured spheres

Let $\overline{\mathcal{Y}}_{n,k}$ and $\overline{\mathcal{Z}}_{n,k}$ be the compactifications of $\mathcal{Y}_{n,k}$ and $\mathcal{Z}_{n,k}$ described in the source, let $\overline{\mathcal{X}}_{n,k}=\overline{\mathcal{Y}}_{n,k}\times\mathbb{A}^k$, and let $\overline{\mathcal{B}}_{n,k}$ be the category of $\mathbb{L}$-graded matrix factorizations of $\mathbf{w}_{n,k}$ on $\overline{\mathcal{X}}_{n,k}$. Equip $V_{n,k}$ with one stop $\Lambda$ corresponding to a point on the component marked by $x_k$. Compactified homological mirror symmetry conjecture. There should exist quasi-equivalences

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

and

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

This is the compactified and partially wrapped extension of the preceding mirror-symmetry conjecture, incorporating a stop on the symplectic side and compactifications on the algebraic side. The source does not provide evidence that these equivalences have been proved.