The Weakly Special Conjecture for potential density

Let $X$ be a smooth projective variety over a number field $K$. A variety is weakly special if no finite étale cover of $X_{\overline{K}}$ rationally dominates a positive-dimensional smooth projective variety of general type over $\overline{K}$. Weakly Special Conjecture. If $X$ is weakly special, then there is a finite extension $L/K$ such that $X(L)$ is dense. This conjecture is presented as the converse to the prediction that dense rational points imply weak specialness; the paper's examples disprove it.