Dual-complex sphere conjecture for log Calabi–Yau pairs

Let $(X,B)$ be an $n$-dimensional log Calabi–Yau pair, and let $\mathcal{D}(X,B)$ denote its dual complex. A finite cover $\pi\colon Y\rightarrow X$ is equipped with the log pull-back boundary $B_Y$. Dual-complex sphere conjecture. There exists a finite cover $\pi\colon Y\rightarrow X$ such that

$$\mathcal{D}(Y,B_Y)\simeq_{\rm PL} S^{n-1}.$$

This predicts that, up to a finite cover, dual complexes of log Calabi–Yau pairs are PL-homeomorphic to spheres. The dual complex is already well-defined up to simple-homotopy equivalence, but the asserted PL-sphere statement remains open.