Lang's strong Diophantine conjecture

Let $V$ be a variety of general type, and let $K$ be a number field.

Lang's strong Diophantine conjecture. There is a proper, closed subset $V_0$ of $V$ such that, for any number field $K$, all but finitely many $K$-rational points of $V$ lie in $V_0$.

The source identifies this as the stronger form due to Lang and says that these Diophantine conjectures are known only for curves and subvarieties of abelian varieties. The general statement remains open.