Green–Griffiths conjecture on pseudo-hyperbolicity

Let $V$ be a variety. It is pseudo-hyperbolic if the locus of images of all nonconstant holomorphic maps $f:{\mathbb C}\longrightarrow V$ is not Zariski dense. A variety is of general type when its canonical divisor has maximal Kodaira dimension.

Green–Griffiths conjecture. If $V$ is of general type, then $V$ is pseudo-hyperbolic.

The conjecture would imply that on a variety of general type the union of curves of genus less than $2$ is not Zariski dense; in particular, a surface of general type would contain only finitely many curves of genus $0$ and $1$. The source mentions progress in dimension $2$, but gives no resolution.