Polyhedrality conjecture for nef section classes

Let $\pi:\mathcal{X}\to B$ be a Fano fibration. Let $A$ be the affine space of curve classes having intersection $1$ with a general fiber, and let $\mathcal{P}$ be the convex hull in $N_1(\mathcal{X})$ of all nef classes in $A$. Polyhedrality conjecture. The set $\mathcal{P}$ is a rational polyhedron whose recession cone is $\operatorname{Nef}_1(\mathcal{X}_\eta)$. In particular, its lattice points are contained in a finite union of translates of $\operatorname{Nef}_1(\mathcal{X}_\eta)_\mathbb{Z}$.

The source states that over a field of characteristic zero this is exactly a proved corollary, while the guiding statement itself is presented generally.