Lang's conjecture on rational points of hyperbolic varieties

Let $Y$ be a projective variety defined over a number field $k$. Write $Y_\mathbb{C}$ for its base change to $\mathbb{C}$, and let $L$ be a number field extending $k$. A complex projective variety is hyperbolic if every holomorphic map $f\colon\mathbb{C}\to Y_\mathbb{C}$ is constant. Lang's conjecture. If $Y_\mathbb{C}$ is hyperbolic, then

$$Y(L)\text{ is finite}$$

for every number field $L$ extending $k$. This conjecture predicts a strong connection between Brody hyperbolicity and arithmetic finiteness of rational points; its status is not established in the supplied source.