Cone theorem for generalized log canonical pairs

Let $(X,B+M)$ be a generalized log canonical pair. The cone of curves is denoted by $\overline{\operatorname{NE}}(X)$, and $\overline{\operatorname{NE}}(X)_{(K_X+B+M)\geq 0}$ is its part on which $K_X+B+M$ is nonnegative.

Cone theorem for g-lc pairs. There are countably many curves $C_j\subset X$ such that $0<-(K_X+B+M)\cdot C_j\leq 2\dim X$ and

$$\overline{\operatorname{NE}}(X)=\overline{\operatorname{NE}}(X)_{(K_X+B+M)\geq0}+\sum\mathbb{R}_{\geq0}[C_j].$$

Moreover, every $(K_X+B+M)$-negative extremal face $F$ has a unique contraction morphism with the stated properties, and numerically trivial line bundles on $F$ descend to the contraction.

This is the expected cone theorem for generalized log canonical pairs and forms part of the anticipated generalized minimal model program. Its status is not resolved in the supplied text.