Campana–Corvaja–Zannier conjecture for special varieties

Let $X$ be a special smooth projective geometrically connected variety over a number field $k$. Campana–Corvaja–Zannier conjecture. There is a number field $L/k$ such that $X_L$ satisfies the weak-Hilbert property over $L$. This is presented as a consequence of Campana's potential-density conjecture and Corvaja–Zannier's weak-Hilbert-property conjecture. It is known for rational varieties and abelian varieties, as well as the additional examples cited by the source; the general statement remains open.