Thin-set version of Manin's conjecture

Let $X$ be a smooth projective variety over a number field $F$ with ample anticanonical class $-K_X$. Let $\mathcal{L}=(L,\|\cdot\|)$ be a big and nef adelically metrized line bundle on $X$. Define the counting function by

$$N(Q,\mathcal L,B)=\#\{x\in Q\mid H_{\mathcal L}(x)\leq B\},$$

and let $a(X,L)$ be the threshold determined by the pseudo-effective cone, while $b(F,X,L)$ is the codimension of its minimal supported face containing $a(X,L)[L]+[K_X]$. A thin subset of $X(F)$ is a finite union of images $\pi_i(Y_i(F))$, where each $\pi_i:Y_i\to X$ is generically finite onto its image and has no rational section. Manin's conjecture. There exists a thin set $Z\subset X(F)$ such that

$$N(X(F)\setminus Z,\mathcal L,B)\sim c(F,X(F)\setminus Z,\mathcal L)B^{a(X,L)}\log(B)^{b(F,X,L)-1},\qquad B\to\infty.$$

Here $c(F,X(F)\setminus Z,\mathcal L)$ is Peyre's constant. This is the thin-set formulation of Manin's conjecture, intended to remove contributions from geometrically incompatible subvarieties and generically finite covers; it remains open in general.