Demailly–Green–Griffiths–Javanpeykar–Kamenova–Lang hyperbolicity equivalence conjecture

Let $X$ be a smooth projective variety over $L$. Consider the four properties: algebraic hyperbolicity over $L$, boundedness over $L$, grouplessness over $L$, and hyperbolicity over $L$. Demailly–Green–Griffiths–Javanpeykar–Kamenova–Lang conjecture. These four properties are equivalent. The source calls this the weak form of the conjectures: implications $(1)\Rightarrow(2)\Rightarrow(3)$ and $(4)\Rightarrow(3)$ are known, as is $(4)\Rightarrow(2)$ by the paper's main theorem; the full equivalence remains open in general, although it is known for closed subvarieties of abelian varieties and for hypersurfaces of sufficiently large degree. Stronger forms concern equivalence of suitable pseudoifications and equality of the associated exceptional loci.