Kawamata–Morrison–Totaro geometric cone conjecture

Let $(X,B)$ be a projective wlc model over $Y$, with $X$ $\mathbb{Q}$-factorial, and assume the relative notation from the arithmetic cone conjecture. A small $\mathbb{Q}$-factorial modification (SQM) for $X/Y$ is a small birational map $\alpha:X\dashrightarrow X^\prime$ over $Y$ with $X^\prime$ projective and $\mathbb{Q}$-factorial over $Y$. Its associated chamber is

$$\textnormal{A}^e(X^\prime/Y)=(\alpha^{-1})_*\textnormal{Nef}^e(X^\prime/Y)\subseteq\textnormal{Mov}^e(X/Y).$$

Geometric cone conjecture. Suppose $(X,B)$ is klt and $X$ is $\mathbb{Q}$-factorial. Then the number of $\textnormal{Aut}(X/Y,B)$-equivalence classes of faces of $\textnormal{Nef}^e(X/Y)$ corresponding to birational contractions or fiber space structures is finite, and the number of $\textnormal{PsAut}(X/Y,B)$-equivalence classes of chambers $\textnormal{A}^e(X^\prime/Y)\subseteq\textnormal{Mov}^e(X/Y)$ corresponding to SQMs $\alpha:X\dashrightarrow X^\prime$ for $X/Y$ is finite.

This is the finiteness, or geometric, counterpart of the cone conjecture: it predicts finitely many contraction faces and movable-cone chambers up to the relevant automorphism actions. The supplied status is unknown.