Uniqueness conjecture for sufficiently positive Manin components

Let $\pi:\mathcal{X}\to B$ be a Fano fibration over an algebraically closed field $\kappa$ of characteristic $0$. Let $A$ be the affine space of section classes and let $\mathcal{P}$ be the convex hull in $N_1(\mathcal{X})$ of $\operatorname{Nef}_1(\mathcal{X})\cap A$. Uniqueness conjecture for Manin components. There is some sufficiently positive $\alpha\in\operatorname{Nef}_1(\mathcal{X}_\eta)$ such that every algebraic equivalence class of curves whose numerical class lies in $\alpha+\mathcal{P}$ is represented by a unique Manin component.

This conjecture simultaneously predicts eventual representability of sufficiently positive classes and uniqueness of the corresponding main component; the source presents it as open.