The strong Green–Griffiths–Lang conjecture for special loci

Let $X$ be a projective complex algebraic variety. Define $\,\operatorname{Sp}(X)$ as the union of the positive-dimensional integral closed subvarieties of $X$ that are not of general type, $\,\operatorname{Sp}_{ab}(X)$ as the union of the images of non-constant rational maps from abelian varieties to $X$, and $\,\operatorname{Sp}_{h}(X)$ as the union of the images of non-constant holomorphic maps from $\mathbb{C}$ to $X$. Strong Green–Griffiths–Lang conjecture. One has

$$\operatorname{Sp}(X)=\operatorname{Sp}_{ab}(X)=\operatorname{Sp}_{h}(X),$$

$\operatorname{Sp}(X)$ is a closed algebraic subvariety of $X$, and $\operatorname{Sp}(X)\neq X$ if and only if $X$ is of general type. Here, for a possibly reducible projective variety, being of general type means that at least one irreducible component is of general type. This is a strong version of conjectures of Green–Griffiths and Lang concerning hyperbolicity and varieties of general type; the paper proves a strong version of the Green–Griffiths–Lang conjecture for projective varieties admitting a large complex local system, but the displayed three-part statement is presented as a conjecture.