Degree of irrationality of a very general Abelian variety

Consider a very general abelian variety of dimension at least and an integer ⁠. We show that if the map has a -dimensional fiber then ⁠. This extends results of the second-named author which covered the cases ⁠. As a geometric application, we prove that any dominant rational map from a very general abelian -fold to has degree at least for ⁠, thus improving results of Alzati and the last-named author in the case of a very general abelian variety.