Bombieri–Lang conjecture on rational points outside a proper subset

Let $X$ be a smooth projective variety defined over a number field $k$. Bombieri–Lang conjecture. There exists a proper Zariski closed subset $\Xi\subsetneqq X$ such that, for every number field extension $k'$ of $k$, the set of $k'$-rational points of $X\setminus\Xi$ is finite. This arithmetic conjecture is cited as an analogy for the generalized Green–Griffiths–Lang conjecture; the source gives no resolution.