Leaps in the depth of compositions of irreducible morphisms

In this article, we give a family of examples of algebras, showing that for every $n\ge 2$ and $m\ge 0$, there is an algebra displaying a path of n irreducible morphisms between indecomposable modules whose composite lies in the $(n+m+3)$-th power of the radical, but not in the $(n+m+4)$-th power. Such an algebra may be also supposed to be string and representation-finite.