For a finite dimensional $k$-Algebra $A$ does infinite representation type imply $(\text{rad}_A^{\omega})^2 \neq 0$?

Question

Let $k$ be a field, $A$ a finite dimensional $k$-Algebra, $\text{mod}\,A$ the category of finite dimensional left $A$-modules and $\text{rad}_A$ the collection of radical morphisms in $\text{mod}\,A$. For a natural number $n$ we denote by $\text{rad}_A^n$ the collection of $n$ compositions of morphisms in $\text{rad}_A$ and set $\text{rad}_A^{\omega} := \bigcap_{n\in \mathbb{N}} \text{rad}_A^n$. It is well-known that if $A$ is of infinite representation type, then $\text{rad}_A^{\omega}\neq 0$. However, in every example it seems to be that we always have $(\text{rad}_A^{\omega})^2 \neq 0$ aswell.

Question: Is it always true that if $A$ is of infinite representation type, then $(\text{rad}_A^{\omega})^2 \neq 0$?

Answer 1

This is proven in the paper (as the title suggests):

Coelho, Flávio U.; Marcos, Eduardo N.; Merklen, Héctor A.; Skowroński, Andrzej, Module categories with infinite radical square zero are of finite type, Commun. Algebra 22, No. 11, 4511-4517 (1994). ZBL0812.16019.