Polyhedrality conjecture for pseudoeffective cones of cycles on k-Fano varieties

Let $X$ be a smooth Fano variety. For an integer $k\geq 1$, call $X$ $k$-Fano if its Chern characters $ch_s(X)$ are positive for $1\leq s\leq k$. Let $\Effc kX$ denote the cone of pseudoeffective $k$-cycles on $X$. Polyhedrality conjecture. If $X$ is $k$-Fano, then $\Effc kX$ is a polyhedral cone. This would generalize Mori's Cone Theorem from curves to higher-dimensional cycles; the paper presents it as a possible generalization, while the provided text gives no resolution.