The relative Green-Griffiths-Lang conjecture

For a proper morphism $f\colon X\to Y$ between two complex algebraic varieties, define the special subsets of $X$ by

$$\operatorname{Sp}_{alg}(X/Y),\quad \operatorname{Sp}_{ab}(X/Y),\quad \operatorname{Sp}_{h}(X/Y).$$

Here $\operatorname{Sp}_{alg}(X/Y)$ is the union of all positive-dimensional irreducible closed subvarieties of $X$ that are not of general type and are contained in a fibre of $f$; $\operatorname{Sp}_{ab}(X/Y)$ is the union of the images of all non-constant rational maps from abelian varieties to $X$ whose images are contained in a fibre; and $\operatorname{Sp}_{h}(X/Y)$ is the union of all entire curves in $X$ contained in a fibre. The morphism $f$ is of general type if, over the generic point of $Y$, its geometric generic fibre is of general type, or more generally if an irreducible component of $X$ is of general type over its image.

Relative Green-Griffiths-Lang conjecture. The following conditions are equivalent:

1. The morphism $f\colon X\to Y$ is of general type.
2. $\operatorname{Sp}_{alg}(X/Y)$ is not Zariski dense in $X$.
3. $\operatorname{Sp}_{ab}(X/Y)$ is not Zariski dense in $X$.
4. $\operatorname{Sp}_{h}(X/Y)$ is not Zariski dense in $X$.

This proposes a relative form of the Green-Griffiths-Lang conjecture, relating the general-type condition on a family to the algebraic, abelian, and entire-curve special loci in its fibres. The source gives no resolution status for the conjecture; its openness and the equivalence of all four conditions remain to be established.