Vojta's height conjecture for log-general-type pairs over function fields

Let $({\mathcal X},{\mathcal D})$ be a pair over a function field $F=k(B)$ whose generic fiber $(X,D)$ is a pair of log general type. For a point $P\in{\mathcal X}(L)$ corresponding to a cover $B_P\to B$ of degree $n$, with field extension $L\supset F$, define

$$\chi(P)=\frac{\chi(B_P)}{n}.$$

Let $N_D^{(1)}(P)$ be the cardinality of the support of $P^*D$.

Vojta's function-field conjecture. For every $\varepsilon>0$ there exist a constant $C$ and a proper closed subvariety $Z$ such that, for all $P\in({\mathcal X}\setminus Z)(\overline F)$,

$$h_{K_{\mathcal X}+D}(P)\leq C\bigl(\chi(P)+N_D^{(1)}(P)\bigr)+O(1).$$

This is the function-field analogue of Vojta's main conjecture for log-general-type pairs. The source presents it as conjectural and relates bounded-height density to isotriviality.