The logarithmic Batyrev–Manin conjecture for integral points

Let $X$ be a variety over a number field $F$, let $D$ be a divisor on $X$, and let $L$ be a big divisor. For a finite set of places $S$ and a model $(\mathcal U,\mathcal L)$ of a dense open subset $U\subset X$, write $N(\mathcal U,\mathcal L,B)$ for the number of $(D,S)$-integral points of $U$ having $\mathcal L$-height at most $B$. Logarithmic Batyrev–Manin conjecture. For any $\epsilon>0$, there exists a dense Zariski open subset $U\subset X$ such that

$$N(\mathcal U,\mathcal L,B)\ll B^{a(L,D)+\epsilon}$$

as $B\rightarrow\infty$. If $-(K_X+D)$ is big, then

$$N(\mathcal U,\mathcal L,B)\gg B^{a(L,D)-\epsilon}$$

as $B\rightarrow\infty$, at least after a suitable finite extension of $F$ and $S$. This is presented as a natural extrapolation of Vojta's conjecture and the Batyrev–Manin conjectures; the asserted upper and lower bounds describe the expected growth of integral points of bounded height, but the source provides no resolution status.