Let $X$ be a Calabi-Yau threefold. From a complex analytic point of view, it is widely believed that it should not be Kobayashi hyperbolic, that is it should always admit some non-constant entire map from the complex plane $f\colon\mathbb C\to X$.

One could be even more ambitious and ask whether a Calabi-Yau threefold always contains a rational or an elliptic curve (or, more generally a non-constant image of a complex torus).

Mostly string theorists have produced lots of examples of such manifolds, mainly by adjunction or crepant resolution of singularities. So my question is:

Is it true that in all known examples of Calabi-Yau threefold one can always find a rational or an elliptic curve (or, more generally a non-constant image of a complex torus)?

Thanks in advance!