Weak Lang-Vojta conjecture for integral points on log-general-type surfaces

Let $X$ be a quasi-projective surface of log-general type, defined over a number field $K$, and let $\mathcal O_S$ be the ring of $S$-integers for a finite set of places of $K$ containing the archimedean ones. Lang-Vojta conjecture. The set

$$X(\mathcal O_S)$$

is not Zariski dense. This is a weaker form of Vojta's conjectures, extending the Bombieri-Lang expectation from rational points on projective varieties of general type to integral points on quasi-projective varieties; it remains open in the stated generality.