On Artin algebras whose indecomposable modules are determined by composition factors

It was conjectured at the end of the book “Representation theory of Artin algebras” by M. Auslander, I. Reiten and S. Smalo that an Artin algebra with the property that its finitely generated indecomposable modules are up to isomorphism completely determined by theirs composition factors is of finite representation type. Examples of rings with this property are the semisimple artinian rings and the rings of the form ${\mathbb{Z}}_n$. An affirmative answer is obtained for some special cases, namely, the commutative, the hereditary and the radical square zero case.