Ducat's log rationality conjecture for three-dimensional Calabi–Yau pairs

Let $(X,B)$ be a pair of dimension three. Assume that $X$ is a rational variety, that

$$K_X+B\sim 0,$$

and that the dual complex satisfies

$$\mathcal{D}(X,B)\simeq_{\rm PL} S^2.$$

Ducat's conjecture. Then the pair $(X,B)$ is log rational. The conjecture extends the demonstrated surface and projective three-space examples, where Cremona transformations relate suitable Calabi–Yau pairs to toric boundaries; its resolution status is not established by the supplied material.