Campana–Corvaja–Zannier conjecture on the weak-Hilbert property

Let $X$ be a smooth proper geometrically connected variety over a finitely generated field $k$ of characteristic zero. Campana–Corvaja–Zannier conjecture. The following are equivalent:

1. There is a finite field extension $L/k$ such that $X_L$ has the weak-Hilbert property over $L$.
2. There is a finite field extension $M/k$ such that $X(M)$ is dense in $X$.
3. The smooth proper connected variety $X_{\overline{k}}$ is special in the sense of Campana.

This conjecture relates the weak-Hilbert property and potential density of rational points to Campana's notion of specialness. The supplied context says that the weak-Hilbert property is conjectured whenever the obvious necessary conditions hold, while the precise status is not given.