Lang's conjecture on entire curves and weakly special manifolds

A projective manifold is weakly special if no finite étale cover of it admits a rational fibration onto a positive-dimensional variety of general type. An entire curve is a holomorphic map from $\mathbf{C}$, and it is Zariski dense if its image is not contained in any proper algebraic subset.

Lang's entire-curve conjecture. (i) A projective manifold of general type does not contain a Zariski dense entire curve. Equivalently, (ii) a projective manifold containing a Zariski dense entire curve is weakly special.

Statement (ii) is presented as an immediate consequence of (i). The conjecture is used as an assumption in the paper and remains open; the source also notes that weakly special manifolds need not contain Zariski dense entire curves.