Strong Lang–Vojta conjecture for exceptional sets

Let $(X,D)$ be a pair defined over a number field $k$, where $X$ is a normal projective variety and $D$ is a normal crossing divisor. The pair is of log general type if, for a log resolution $(Y,D_Y)$, the divisor $K_Y+D_Y$ is big. An $(S,D)$-integral point is a section of a model of $(X,D)$ over $\operatorname{Spec} \mathcal O_k$ whose pulled-back boundary is supported in $S$.

Strong Lang–Vojta conjecture. The pair $(X,D)$ is of log general type if and only if there exists a proper closed subset

$$Z=\operatorname{Exc}(X,D),$$

called the exceptional set, such that $X\setminus Z$ has only finitely many $(S',D)$-integral points for every finite extension $k'\supset k$, where $S'$ is the set of places above $S$.

This conjecture predicts that the arithmetic of the complement of the exceptional set is finite precisely for pairs of log general type. It is presented as a fundamental conjecture governing rational and integral points, and the supplied text gives no resolution status.