The Gabriel-Roiter submodules of simple homogeneous modules

Let $\Lambda$ be a connected tame hereditary algebra over an algebraically closed field. We show that if $\Lambda=kQ$ is of type $\widetilde{\mathbb{A}}_n$, $\widetilde{\mathbb{D}}_n$, $\widetilde{\mathbb{E}}_6$ or $\widetilde{\mathbb{E}}_7$, then every Gabriel-Roiter submodule of a quasi-simple module of rank $1$ (i.e. a simple homogeneous module) has defect $-1$. In particular, any Gabriel-Roiter submodule of a simple homogeneous module yields a Kronecker pair, and thus induces a full exact embedding of the category $\mod k\widetilde{\mathbb{A}}_1$ into $\mod\Lambda$, where $\widetilde{\mathbb{A}}_1$ is the Kronecker quiver. Consequently, we obtain that all quasi-simple modules are Gabriel-Roiter factor modules.