Algebraic Green–Griffiths–Lang conjecture for log varieties

Let $X$ be a smooth projective variety and $D$ a simple normal crossings divisor such that $K_X+D$ is big. Let $L$ be an ample line bundle on $X$, and let $g(C)$ denote the geometric genus of a smooth projective curve $C$. Algebraic Green–Griffiths–Lang conjecture. There exist an $\epsilon>0$ and a subvariety $S\subsetneq X$ such that, for every nonconstant map $f:C\to X$ from a smooth projective curve satisfying $f(C)\not\subset D\cup S$, one has

$$2g(C)-2+|f^{-1}(D)|>\epsilon\deg f^*L.$$

The source identifies this as Demailly's weaker algebraic version of the Green–Griffiths–Lang conjecture and notes that it is more tractable than the full analytic statement; the paper proves it for the complements under study.