Generalized Green-Griffiths-Lang conjecture for foliated manifolds

Let $(X,\mathcal{F})$ be a projective foliated manifold, where $\operatorname{Sing}\mathcal{F}$ consists only of canonical singularities and $K_{\mathcal{F}}$ is big. A non-degenerate holomorphic map is a holomorphic map $f:\mathbb{C}^p\to X$ tangent to $\mathcal{F}$ whose image is not contained in a proper subvariety of the relevant leaf space.

Generalized Green-Griffiths-Lang conjecture. There exists an algebraic subvariety $Y\subsetneq X$ such that any non-degenerate holomorphic map

$$f:\mathbb{C}^p\to X$$

tangent to $\mathcal{F}$ has image

$$f(\mathbb{C}^p)\subset Y.$$

This extends the Green-Griffiths-Lang principle to holomorphic maps tangent to foliations with canonical singularities and big canonical bundle. The preceding theorem establishes the analogous containment in the logarithmic simple singularity case; the stated generalization remains unresolved in the source.