Independence of irreducibility from the classical-group representation

Let $G_n$ be a classical group, let $\tau$ be an irreducible representation of $G_n$, and let $\pi\in\operatorname{Irr}\operatorname{GL}$ satisfy

$$\operatorname{supp}\pi\cap S_\tau=\emptyset.$$

Irreducibility-independence conjecture. The irreducibility of

$$\pi\rtimes\tau$$

depends only on $\pi$ and not on $\tau$. This would extend the preceding reducibility results beyond their additional hypotheses; the paper explicitly states that the authors do not know whether the corresponding necessary condition holds in general.