Kawamata–Morrison–Totaro arithmetic cone conjecture

Let $(X,B)$ be a fixed projective wlc model of dimension $d\in\mathbb{Z}_{\ge 2}$ over $Y$, with $K_X+B\sim_{\mathbb{R},Y}0$, and assume abundance for $(X,B)$ and all its projective wlc models. Let $N^1_{\mathbb{R}}(X/Y)$ be the relative Néron–Severi space, and let $\textnormal{Nef}^e(X/Y)$ and $\overline{\textnormal{Mov}}^e(X/Y)$ denote the effective nef and effective closed movable cones. Let $\textnormal{Nef}^+(X/Y)$ and $\overline{\textnormal{Mov}}^+(X/Y)$ be their rational convex hulls. Here $\textnormal{Aut}(X/Y,B)$ and $\textnormal{PsAut}(X/Y,B)$ act by pullback on $N^1_{\mathbb{R}}(X/Y)$.

Arithmetic cone conjecture. Suppose $(X,B)$ is klt. Then there is a rational polyhedral cone $\Pi\subseteq\textnormal{Nef}^e(X/Y)$ such that

$$\textnormal{Aut}(X/Y, B) \cdot \Pi = \textnormal{Nef}^e(X/Y) = \textnormal{Nef}^+(X/Y),$$

and $\operatorname{Int}(\Pi)\cap\operatorname{Int}(\varphi^*\Pi)=\emptyset$ for every $\varphi\in\textnormal{Aut}(X,B)$ unless $\varphi^*=\operatorname{id}$. There is also a rational polyhedral cone $\Pi^\prime\subseteq\overline{\textnormal{Mov}}^e(X/Y)$ such that

$$\textnormal{PsAut}(X/Y, B) \cdot \Pi^\prime = \overline{\textnormal{Mov}}^e(X/Y) = \overline{\textnormal{Mov}}^+(X/Y),$$

and $\operatorname{Int}(\Pi^\prime)\cap\operatorname{Int}(\varphi^*\Pi^\prime)=\emptyset$ for every $\varphi\in\textnormal{PsAut}(X,B)$ unless $\varphi^*=\operatorname{id}$.

This is the cone conjecture in its nef and movable forms: the relevant cones should admit rational polyhedral fundamental domains for the automorphism and pseudo-automorphism actions. The source refers to recent progress and an overview, but the supplied status is unknown.