Morrison's cone conjecture for Calabi–Yau threefolds

Let $X$ be a Calabi–Yau $3$-fold. The automorphism group of $X$ acts on the nef cone of $X$ with a rational polyhedral fundamental domain, and the pseudoautomorphism group of $X$ acts on the movable cone of $X$ with a rational polyhedral fundamental domain.

Morrison's cone conjecture.

$$\begin{aligned} &\text{The automorphism group of }X\text{ acts on the nef cone of }X\text{ with a rational polyhedral fundamental domain; and}\\ &\text{the pseudoautomorphism group of }X\text{ acts on the movable cone of }X\text{ with a rational polyhedral fundamental domain.} \end{aligned}$$

The conjecture predicts a finite polyhedral description of the nef and movable cones up to automorphisms and pseudoautomorphisms. The paper's abstract states that the relevant special case is confirmed under the hypothesis $H^3(Y,\mathbb{C})=0$, while the general conjecture remains open.