Green-Griffiths-Lang conjecture for log-general-type varieties

Let $X$ be a smooth complex projective variety, and let $D$ be a normal crossings divisor on $X$. Suppose that $X\setminus D$ is a variety of log-general type.

Green-Griffiths-Lang conjecture. Every entire curve $f: \mathbb{C} \to X\setminus D$ is algebraically degenerate; that is, the image of $f$ is contained in a proper algebraic subvariety of $X$. Furthermore, there exists a proper algebraic subvariety $Z \subset X$ such that the image of every entire curve $f: \mathbb{C} \to X$ is contained in $Z$.

This conjecture predicts the hyperbolic behavior of entire curves on quasi-projective varieties of log-general type. The source presents it as the motivating conjecture for its results on hyperbolicity and algebraic degeneracy; no resolution status is supplied here.