Sphere conjecture for coregularity-zero Fano type varieties

Let $X$ be an $n$-dimensional Fano type variety of coregularity zero. A $2$-complement $(X,B)$ is a complement with $(X,B)$ log canonical and $2(K_X+B)\sim 0$. Let $(Y,B_Y)$ be the log pull-back of $(X,B)$ to a finite cover $Y\rightarrow X$. Sphere conjecture for coregularity-zero Fano type varieties. There exists a $2$-complement $(X,B)$ and a finite cover $Y\rightarrow X$ such that $(Y,B_Y)$ is a log Calabi–Yau pair and

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

This predicts that coregularity-zero Fano type varieties admit complements whose dual complexes become spheres after finite covers, extending the toric picture. The source presents this as an open conjecture.