Algebro-geometric Poincaré conjecture for dual complexes

Let $(X,B)$ be an $n$-dimensional 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 $\mathcal{D}(X,B)$ be its dual complex, let $S^k$ be the $k$-sphere, and let $O(k)$ be the orthogonal group. Algebro-geometric Poincaré conjecture. There exist an integer $k\leq n-1$ and a finite subgroup $G\leq O(k)$ such that

$$\mathcal{D}(X,B)\simeq_{\rm PL} S^k/G.$$

This folklore conjecture was stated as a question by Kollár and Xu and is motivated by viewing dual complexes as combinatorial counterparts of spherical local models; no resolution is supplied here.