Lang's hyperbolicity and Mordellicity conjectures

Let $X$ be a projective variety. A projective variety is mordellic if $X(K)$ is finite for every field $K$ finitely generated over $\mathbb{Q}$. For the second assertion, define

$$\operatorname{Exc}(X)=\bigcup \overline{\{f(\mathbb{C}):f:\mathbb{C}\to X\}}.$$

Lang's conjectures. If $X$ is hyperbolic, then it is mordellic. Moreover, $X\setminus\operatorname{Exc}(X)$ is mordellic.

These assertions seek a common geometric and arithmetic analogue of the Green–Griffiths phenomenon. The source attributes them to Lang and gives no resolution.