Weakly special potential-density conjecture

Let $X$ be a weakly special variety over a number field $K$. Here potentially dense integral points means that, after a finite extension of $K$ and enlarging a finite set of places, the integral points of a finite-type model are dense in $X$.

Weakly Special Conjecture. If $X$ is a weakly special variety over a number field $K$, then $X$ has a potentially dense set of integral points.

The conjecture is presented as the converse direction to the Lang–Vojta–Chevalley–Weil prediction. It is known in several cases, including abelian varieties, Enriques surfaces, and certain K3 surfaces and Calabi–Yau threefolds, but the source expects it to fail for some weakly special varieties.