Weak Manin's conjecture for rational point counting

Let $X$ be a geometrically uniruled smooth projective variety defined over a number field $k$, let $L$ be a big and nef divisor on $X$, and let

$$H_L:X(k)\rightarrow\mathbb R_{>0}$$

be a height function. For a suitable Zariski open subset $U\subset X$, define

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

The $a$-invariant is

$$a(X,L)=\inf\{t\in\mathbb R\mid K_X+tL\in\overline{\operatorname{Eff}}^1(X)\},$$

where $\overline{\operatorname{Eff}}^1(X)$ is the cone of pseudo-effective divisors on $X$. Weak Manin's conjecture. There exists a non-empty Zariski open subset $U\subset X$ such that, for every $\epsilon>0$,

$$N(U,L,T)=O_\epsilon\left(T^{a(X,L)+\epsilon}\right).$$

For $L=-K_X$, one has $a(X,L)=1$, explaining the terminology “linear growth”. The paper notes that the analogous assertion with $L$ merely big, rather than big and nef, has counterexamples; the conjecture is presented as a weaker form of Manin's conjecture.