Demailly–Green–Griffiths–Lang–Vojta equivalence conjecture

Let $X$ be a projective variety over a field $k$ of characteristic zero. The properties used below are algebraic hyperbolicity, boundedness, $(n,m)$-boundedness for integers $n,m\geq 1$, having every integral closed subvariety of general type, and grouplessness.

Demailly–Green–Griffiths–Lang–Vojta conjecture. The following statements are equivalent: $X$ is algebraically hyperbolic over $k$; $X$ is bounded over $k$; for all $n\geq 1$ and $m\geq 1$, $X$ is $(n,m)$-bounded over $k$; every integral closed subvariety of $X$ is of general type; and $X$ is groupless over $k$.

This conjecture unifies several expected characterisations of hyperbolicity and boundedness for projective varieties. The source presents it as evidence for conjectures of Demailly, Green–Griffiths, Lang, and Vojta, but gives no resolution in the stated generality.