Weak Lang–Vojta conjecture I

Let $k$ be an algebraically closed field of characteristic zero. A variety $X$ over $k$ is strongly-Brody hyperbolic over $k$ if, for every subfield $k_0\subset k$, every model $\mathcal{X}$ for $X$ over $k_0$, and every embedding $k_0\to\mathbb{C}$, the variety $\mathcal{X}_{\mathbb{C}}$ is Brody hyperbolic. Weak Lang–Vojta conjecture I. If $X$ is a projective variety over $k$, then $X$ is Mordellic over $k$ if and only if $X$ is strongly-Brody hyperbolic over $k$. This is the first weak form of the Lang–Vojta conjectures; the source provides no resolution status.