Abelian bordered factors and periodicity

A finite word is bordered if it has a non-empty proper prefix which is equal to its suffix, and unbordered otherwise. Ehrenfeucht and Silberger proved that an infinite word is (purely) periodic if and only if it contains only finitely many unbordered factors. We are interested in an abelian modification of this fact. Namely, we have the following question: Let w be an infinite
word such that all sufficiently long factors are abelian bordered. Is w (abelian) periodic? We also consider a weakly abelian modification of this question, when only the frequencies of letters are taken into account. Besides that, we answer a question of Avgustinovich, Karhumaki and Puzynina concerning abelian central factorization theorem.