Gaitsgory–Kazhdan irreducibility conjecture for induced representations

Let $G$ be a split reductive group over a local field $\bK$, let $\BG=G((t))$ be the corresponding loop group, and let $\pi$ be an irreducible cuspidal representation of $\bG$. Let $i^\BG_\bG$ be the induction functor and set

$$\Pi:=i^\BG_\bG(\pi)\in \operatorname{Rep}(\BG).$$

Gaitsgory–Kazhdan irreducibility conjecture. The object $\Pi\in \operatorname{Rep}(\BG)$ is irreducible.

This is the loop-group analogue of the irreducibility of representations induced from cuspidal representations of reductive groups. The source presents it as a formulation of Conjecture 4.7 from the authors' earlier work; the paper proves the relevant endomorphism calculation but does not establish irreducibility in general.