Thinness conjecture for geometrically incompatible covers

Let $X$ be a smooth uniruled variety over a number field $F$ and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. For every $F$-morphism $f:Y\to X$ from a smooth projective variety $Y$, assume that $f$ is generically finite onto its image. Let $a(X,L)$ and $b(F,X,L)$ be the associated geometric invariants. Thinness conjecture for incompatible covers. As $f$ varies over all such morphisms for which either $f^*L$ is not big or

$$(a(Y,f^*L),b(F,Y,f^*L))>(a(X,L),b(F,X,L))$$

in the lexicographic order, the set

$$\bigcup_f f(Y(F))$$

is contained in a thin subset of $X(F)$. This conjecture asserts that all contributions violating the expected Manin growth data can be removed simultaneously by a thin set; it is presented as the mechanism preventing these geometric incompatibilities from obstructing the thin-set version of Manin's conjecture.