Totaro's cone conjecture for klt Calabi–Yau pairs

Let $(X,\Delta)$ be a $\mathbb{Q}$-factorial projective klt Calabi–Yau pair, meaning that $X$ is normal, $\Delta$ is an effective $\mathbb{R}$-divisor, $K_X+\Delta$ is $\mathbb{R}$-Cartier, and $K_X+\Delta$ is numerically trivial. Let $\operatorname{Nef^e}(X)=\operatorname{Nef}(X)\cap\operatorname{Eff}(X)$ be the effective nef cone, and let $\operatorname{\overline{Mov}^e}(X)=\operatorname{\overline{Mov}}(X)\cap\operatorname{Eff}(X)$ be the effective movable cone.

Totaro's cone conjecture. There exists a rational polyhedral cone $\Pi$ that is a fundamental domain for the action of $\operatorname{Aut}(X)$ on $\operatorname{Nef^e}(X)$, namely

$$\operatorname{Nef^e}(X)=\bigcup_{g\in\operatorname{Aut}(X)}g^*\Pi,$$

with $\operatorname{Int}\Pi\cap\operatorname{Int}g^*\Pi=\emptyset$ unless $g^*=\operatorname{id}$. There also exists a rational polyhedral cone $\Pi'$ that is a fundamental domain for the action of $\operatorname{PsAut}(X)$ on $\operatorname{\overline{Mov}^e}(X)$.

This is a generalized Morrison–Kawamata cone conjecture for klt Calabi–Yau pairs, concerning the effective nef and movable cones rather than the full cones. The source attributes the formulation to Totaro and does not state a resolution here.