The Ext criterion for irreducibility of parabolic induction

Let $C,D\in\mathbf{Comp}$, and let $\pi(C)$ and $\pi(D)$ be the irreducible representations corresponding to these components. For points $x\in C$ and $y\in D$, write $\operatorname{Ext}^1_\Pi(x,y)$ for the relevant first Ext group.

Ext criterion conjecture. The representation $\pi(C)\times\pi(D)$ is irreducible if and only if there exist open subsets $U_C\subseteq C$ and $U_D\subseteq D$ such that

$$\dim_\mathbb{C}\operatorname{Ext}^1_\Pi(x,y)=0$$

for all $(x,y)\in U_C\times U_D$.

This conjecture proposes that irreducibility of parabolic induction is governed by generic vanishing of the first Ext group. The source attributes it to Geiss–Schröer and Lapid–Mínguez.