Geometric Lang–Vojta conjecture for integral points on log-general-type pairs

Let $k$ be an algebraically closed field, let $B$ be a nonsingular irreducible algebraic curve over $k$ with function field $K=k(B)$, and let $(X,D)$ be a pair of log general type over $K$, where $X$ is a smooth integral projective $K$-variety and $D\subset X$ is a fixed reduced effective divisor. Let $(\mathcal X,\mathcal D)$ be a model of $(X,D)$ over $B$, and let $S\subset B$ be finite. Writing

$$\mathcal O_{S,\mathcal D}(\mathcal X):=\{\sigma_P\in X(K): f(\sigma_P(B)\cap\mathcal D)\subseteq S\}$$

for the set of $(S,\mathcal D)$-integral sections, and letting $\mathcal Z$ denote the Zariski closure in $\mathcal X$ of a closed subset $Z\subset X\smallsetminus D$, Geometric Lang–Vojta conjecture. There exist a proper closed subset $Z\subset X\smallsetminus D$ and a constant $m=m(\mathcal X,\mathcal D)>0$ such that, for every $\sigma_P\in\mathcal O_{S,\mathcal D}(\mathcal X)$ with $\sigma_P(B)\not\subset\mathcal Z$,

$$\deg_{B}\sigma_P^*\mathcal D\leq m\,(2g(B)-2+\# S).$$

This conjecture predicts uniform bounds on the intersection of integral sections with the boundary divisor for pairs of log general type. It is known in some cases, but for a general complex abelian variety—particularly when its $\operatorname{Tr}_{K/\mathbb C}(A)\neq 0$ and $A$ is nonconstant—the conjecture remains open.