Irreducibility conjecture for semi-infinite inductions from cuspidal representations

Let $G$ be the group under consideration, let $\widehat{G}$ denote its associated central extension, and let $\Pi$ be an irreducible cuspidal representation of $G$. The functor $i^{\widehat{G}}_G$ produces a representation $i^{\widehat{G}}_G(\Pi)$ of $\widehat{G}$. Irreducibility conjecture. The objects $i^{\widehat{G}}_G(\Pi)$, for $\Pi$ being a cuspidal representation of $G$, are irreducible. This is proposed by analogy with the corresponding irreducibility result for cuspidal representations of $p$-adic groups; the paper gives evidence but does not establish the conjecture.