The finite-quotient sphere conjecture for dual complexes

Let $(X,B)$ be a log Calabi–Yau pair, meaning that $X$ is a proper variety, $B$ is an effective $\mathbb{R}$-divisor, $(X,B)$ is log canonical, and $K_X+B$ is $\mathbb{R}$-linearly trivial. Let $\mathcal{D}(X,B)$ denote its dual complex. Finite-quotient sphere conjecture. The dual complex $\mathcal{D}(X,B)$ is homeomorphic to a finite quotient of a sphere. This is presented as a generalization of the expected spherical local models in mirror symmetry and as a question previously posed by Kollár and Xu; its resolution is not indicated here.