Theta representation as the least generic representation

Let $\overline{G}$ be an $n$-fold persistent covering group. Let $\Theta(\overline{G},\chi)=\pi_{\Gamma^-}$ be the theta representation associated to an unramified character $\chi$ satisfying $\Phi(\chi)=\Delta$. Theta least-genericity conjecture. The representation $\Theta(\overline{G},\chi)$ is the least generic representation among all irreducible genuine representations with the same Bernstein support: for every irreducible constituent $\pi$ of an arbitrary unramified genuine principal series of $\overline{G}$,

$$\dim \operatorname{Wh}_\psi(\pi)\geq\dim \operatorname{Wh}_\psi(\Theta(\overline{G},\chi)).$$

This predicts the covering-group analogue of the standard module conjecture. The source presents it as a belief and gives no resolution.