Generic vanishing index and the birationality of the bicanonical map of irregular varieties

We prove that any smooth complex projective variety with generic vanishing index bigger or equal than 2 has birational bicanonical map. Therefore, if X is a smooth complex projective variety φ with maximal Albanese dimension and non-birational bicanonical map, then the Albanese image of X is fibred by subvarieties of codimension at most 1 of an abelian subvariety of Alb X.