Mirror-symmetry compactification conjecture for genus-zero relative character varieties

Let $n>3$ and let $\boldsymbol{t}\bigl(=\mf{t}\bigr)\in\mathbb{C}^n$ be generic. Let $\mathcal{R}_{n,\boldsymbol{t}}$ be the genus-zero $n$-punctured relative character variety. A smooth compactification is a smooth variety $\overline{\mathcal{R}}_{n,\boldsymbol{t}}$ containing $\mathcal{R}_{n,\boldsymbol{t}}$ as an open subset, with boundary divisor $\mathcal{D}_{n,\boldsymbol{t}}=\overline{\mathcal{R}}_{n,\boldsymbol{t}}\setminus\mathcal{R}_{n,\boldsymbol{t}}$. The dual intersection complex $\mathcal{K}$ of $\mathcal{D}_{n,\boldsymbol{t}}$ records the incidences among the components of this divisor. The pair $(\overline{\mathcal{R}}_{n,\boldsymbol{t}},\mathcal{D}_{n,\boldsymbol{t}})$ is log Calabi–Yau when its log canonical class is trivial.

Mirror-symmetry compactification conjecture. For every $n>3$ and generic $\boldsymbol{t}\in\mathbb{C}^n$, there exists a smooth compactification $\overline{\mathcal{R}}_{n,\boldsymbol{t}}$ such that $\mathcal{D}_{n,\boldsymbol{t}}\subset\overline{\mathcal{R}}_{n,\boldsymbol{t}}$ is a reduced simple normal crossings divisor, its dual intersection complex $\mathcal{K}$ is homeomorphic to

$$S^{2(n-3)-1},$$

and $(\overline{\mathcal{R}}_{n,\boldsymbol{t}},\mathcal{D}_{n,\boldsymbol{t}})$ is a log Calabi–Yau pair.

Such compactifications are expected to provide the B-side mirror family for the genus-zero relative character varieties and to organize their connections with Hitchin moduli spaces, skein algebras, cluster varieties, and Gromov–Witten constructions. The source presents this as an expected property and gives no evidence of a resolution; the parenthetical remark indicates that $\mathbb{Q}$-factorial divisorial log terminal singularities may suffice for some applications, but the stated conjecture requires smoothness.