The Weak Specialness Conjecture for rational points

Let $X$ be a projective variety defined over a number field. A variety is weakly special if some (equivalently, any) desingularization admits no dominant rational map to a positive-dimensional variety of general type after any finite étale cover. A set of rational points is potentially dense if it becomes Zariski dense after passing to a finite extension of the number field. Weak Specialness Conjecture. The set of rational points on $X$ is potentially dense if and only if $X$ is weakly special.

This conjecture proposes that weak specialness characterizes potential density for projective varieties over number fields. It remains open; in particular, there is no known example of a weakly special variety whose rational points are not potentially dense.