Lang's conjecture relating entire curves and integral sets

Let $X$ be an algebraic variety defined over a number field $K$. A holomorphic map $f:\mathbb{C}\to X(\mathbb{C})$ is non-constant if it is not constant, and its image is Zariski dense if its Zariski closure is all of $X(\mathbb{C})$. An integral set on $X$ means a set of integral points in the usual arithmetic sense.

Lang's conjecture. There exists a non-constant holomorphic map

$$f:\mathbb{C}\to X(\mathbb{C})$$

with Zariski dense image if and only if there is a finite field extension $K'/K$ such that $X$ admits an infinite, respectively Zariski dense, integral set.

This conjecture expresses the proposed correspondence between the diophantine behavior of varieties over number fields and their complex-analytic properties, with entire curves corresponding to infinite sets of integral points for non-compact varieties. The source presents it as part of a philosophy widely conjectured by Serge Lang and others; its resolution is not specified here.