Weak Manin's conjecture for geometrically uniruled varieties

Let $F$ be a number field, let $X$ be a smooth geometrically uniruled projective variety over $F$, and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. Choose a height function associated to $L$,

$$\mathsf H_L:X(F)\to\mathbb R_{\geq 0}.$$

For $T\geq 0$, define

$$N(U,L,T)=\#\{P\in U(F)\mid\mathsf H_L(P)\leq T\}.$$

Weak Manin's conjecture. There exists a non-empty Zariski open subset $U\subset X$ such that, for every $\epsilon>0$,

$$N(U,L,T)=O\bigl(T^{a(X,L)+\epsilon}\bigr).$$

This is the weak upper-bound form of Manin's conjecture for rational points on geometrically uniruled varieties. It predicts an upper bound with exponent arbitrarily close to the $a$-invariant; the supplied text gives no evidence that the conjecture has been resolved.