Green–Griffiths–Lang conjecture for Mordellic varieties

Let $k$ be a number field and let $k^{\mathrm{alg}}$ be an algebraic closure of $k$. Let $X$ be a smooth projective geometrically irreducible variety over $k$. The properties of $X$ being Mordellic, every rational map from every abelian variety over $k^{\mathrm{alg}}$ to $X_{k^{\mathrm{alg}}}$ being constant, and every complex analytification $X_\sigma^{\mathrm{an}}$ being Brody hyperbolic are the conditions in question. Green–Griffiths–Lang conjecture. The following are equivalent:

$$\begin{aligned} &X\text{ is Mordellic};\\ &\text{for every abelian variety }A\text{ over }k^{\mathrm{alg}},\text{ all rational maps }f:A\dashrightarrow X_{k^{\mathrm{alg}}}\text{ are constant};\\ &\text{for every embedding }\sigma:k\to\mathbb{C},\ X_\sigma^{\mathrm{an}}\text{ is Brody hyperbolic,}\text{ i.e. every holomorphic map }f:\mathbb{C}\to X_\sigma^{\mathrm{an}}\text{ is constant.} \end{aligned}$$

This conjecture seeks equivalent arithmetic, algebraic, and complex-analytic characterizations of varieties with finite rational-point sets over every finite extension. Its general status is open.