Strong Lang–Vojta conjecture III

Let $k$ be an algebraically closed field of characteristic zero. A variety over $k$ is strongly-pseudo-Brody hyperbolic (respectively, strongly-pseudo-Kobayashi hyperbolic) if every model over every subfield of $k$ has the corresponding pseudo-hyperbolic complex realization under every embedding into $\mathbb{C}$. Strong Lang–Vojta conjecture III. For a projective variety $X$ over $k$, strong pseudo-Brody hyperbolicity, strong pseudo-Kobayashi hyperbolicity, pseudo-Mordellicity, pseudo-arithmetic hyperbolicity, pseudo-grouplessness, and being of general type are equivalent. The source explicitly says that the conjecture remains unresolved in general.