On simple-minded systems and $tau$-periodic modules of self-injective algebras

Let $A$ be a finite-dimensional self-injective algebra over an algebraically closed field, $mathcal{C}$ a stably quasi-serial component (i.e. its stable part is a tube) of rank $n$ of the Auslander-Reiten quiver of $A$, and $mathcal{S}$ be a simple-minded system of the stable module category $stmod{A}$. We show that the intersection $mathcal{S}capmathcal{C}$ is of size strictly less than $n$, and consists only of modules with quasi-length strictly less than $n$. In particular, all modules in the homogeneous tubes of the Auslander-Reiten quiver of $A$ cannot be in any simple-minded system.