Rational dlt strata and spherical dual complex imply birational toricity

Let $(X,B)$ be a log Calabi--Yau pair of dimension $n$. Write $4\mathcal{D}(X,B)4$ for its dual complex, and let $c_{\rm bir}(X,B)$ denote its birational complexity. Let $\Sigma^n$ be the sum of the hyperplane coordinates of $\mathbb{P}^n$. Assume

$$\mathcal{D}(X,B)\simeq_{\rm PL} S^{n-1}$$

and that every dlt stratum of $(X,B)$ is rational. Birational toricity conjecture. Then

$$c_{\rm bir}(X,B)=0.$$

In particular, there is a crepant birational map

$$(\mathbb{P}^n,\Sigma^n)\dashrightarrow (X,B).$$

This would characterize a broad class of log Calabi--Yau pairs admitting birational toric models. The source presents the assertion as a fundamental conjecture, and no resolution is supplied here.