Geiss–Schröer rigidity conjecture for irreducibility of induced representations

Let $C_i\in\operatorname{Comp}(\bm{d}_i)$ for $i=1,2$, and suppose that at least one of $C_1,C_2$ is rigid, meaning that it contains an open orbit. Let $\pi(C_i)$ be the corresponding irreducible representations, and let $C_1$ and $C_2$ commute in the component-theoretic sense. Geiss–Schröer's rigidity conjecture.

$$\pi(C_1)\times\pi(C_2)\text{ is irreducible}\quad\Longleftrightarrow\quad C_1\text{ and }C_2\text{ commute}.$$

The paper records this as a specialization of the preceding Geiss–Schröer conjecture and notes that the conjecture is wide open in general.