Lang–Vojta–Chevalley–Weil weakly special conjecture

Let $X$ be a variety over a number field. A variety is weakly special if it satisfies the geometric condition defined in Campana's theory; a variety satisfies potential density of integral points if, after a finite extension of the ground number field and removal of finitely many places, its integral points are dense.

Lang–Vojta–Chevalley–Weil conjecture. If $X$ satisfies potential density of integral points, then $X$ is weakly special.

This is motivated by the Lang–Vojta prediction that integral points on varieties of log-general type are not dense, together with Chevalley–Weil. The source gives no resolution of this conjecture.