The abelian-variety entire-curve and Kobayashi-distance conjecture

Let $A$ be a complex abelian variety, let $X$ be a subvariety of $A$, and let $E$ be the union of all translates of complex subtori of $A$ that are contained in $X$.

Abelian-variety entire-curve conjecture. Every non-constant holomorphic map

$$\nf:\mathbb C\to X$$

\nhas image contained in $E$, and the Kobayashi pseudodistance on $X$ is a distance outside $E$.

The claim extends the known fact that entire maps to $X$ with bounded derivative have image in $E$, since such maps to an abelian variety are induced by affine-linear maps. The source gives no resolution of either assertion, so the conjecture remains open.