Strong Manin's conjecture on uniqueness of Manin components

Let $X$ be a smooth projective uniruled variety and let $L$ be a big and nef $\mathbb{Q}$-divisor such that $\kappa(K_X+a(X,L)L)=0$. Let $F(X,L)$ be the relevant face of the nef cone of curve classes, and let a Manin component mean a component satisfying the four conditions defined in the paper: its classes lie in $F(X,L)$, and its family map does not factor through the specified higher-$a$, face-contracting, or positive-Iitaka-dimension thin morphisms.

Strong Manin's conjecture. For any $\mathbb{Z}$-curve class $\alpha$ contained in the relative interior of $F(X,L)$, there is at most one Manin component parametrizing curves of class $\alpha$.

This is the paper's main conjecture concerning Manin components. Uniqueness would give a particularly precise form of the expected geometric analogue of Manin's conjecture; the source says it holds in the examples known there and verifies it for the stated toric situation.