Lehmann–Tanimoto's geometric consistency conjecture for Manin's conjecture

Let $X$ be a smooth projective uniruled variety over a number field $F$. Let $L$ be the line bundle used to define the Fujita invariant $a(X,L)$ and the $b$-constant. For a smooth projective variety $Y$ and a generically finite $F$-morphism $f:Y\to X$, compare the pairs $(a(Y,f^*L),b(Y,f^*L))$ and $(a(X,L),b(X,L))$ in lexicographic order.

Lehmann–Tanimoto's conjecture. Considering all such morphisms $f$ for which $f^*L$ is not big or

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

the union of their rational images satisfies

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

for some thin subset $Z\subset X$.

This conjecture asserts that generically finite covers with geometrically incompatible invariants do not obstruct the thin-set version of Manin's conjecture. It is attributed in the source to Lehmann and Tanimoto and is presented as an open conjecture.