Strong Green–Griffiths conjecture on a uniform exceptional subvariety

Let $X$ be a projective complex variety of general type. A closed subvariety $E\subsetneq X$ is a proper closed subvariety of $X$.

Strong Green–Griffiths conjecture. There exists a closed subvariety $E\subsetneq X$ such that $E$ contains the image $f(\mathbf{C})$ for every non-constant holomorphic map

$$f:\mathbf{C}\to X.$$

This strengthens the Green–Griffiths conjecture by requiring a single proper subvariety to contain every non-constant entire curve, rather than allowing the exceptional subvariety to depend on the map. The source introduces this as a strengthening; no resolution is given there.