Wilson's weaker cone conjecture for Calabi–Yau threefolds

Let $Z$ be a Calabi–Yau threefold with Kähler cone $k_Z$ and closure $\overline{k}_Z$, and let $c_2(Z)$ denote its second Chern class. Say that $c_2$ is strictly positive on $\overline{k}_Z$ when it has positive intersection with every nonzero class in the closure of the Kähler cone.

Wilson's weaker cone conjecture. If $c_2$ is strictly positive on the closure of the Kähler cone of a Calabi–Yau threefold, then the Kähler cone of $Z$ is generated by finitely many rational extremal rays; equivalently, it is rational polyhedral.

This is presented as a weaker version of Morrison's conjecture, motivated by Wilson's finiteness theorem for the automorphism group under the same positivity hypothesis. The source does not state a resolution of this conjecture.