The non-genericity classification conjecture for unramified metaplectic representations

Let $G^{(r)}$ be an $r$-fold metaplectic covering group, and let $\pi$ be the unramified representation of $G^{(r)}$ defined in the preceding construction. Suppose that $H$ is contained in a Levi part of a parabolic subgroup of $G$, and that $GL_1^k\times H$ is a Levi part of a parabolic subgroup $P$ of $G$. For an arbitrary unramified character $\mu$ of $GL_1^k$, let $\Theta_H^{(r)}$ be the Theta representation of $H^{(r)}$, and consider

$$\operatorname{Ind}_{P^{(r)}}^{G^{(r)}}\mu\Theta_H^{(r)}\delta_P^{1/2}.$$

Non-genericity classification conjecture. Ignoring the special cases mentioned in the paper, $\pi$ is not generic if and only if it is a subrepresentation of

$$\operatorname{Ind}_{P^{(r)}}^{G^{(r)}}\mu\Theta_H^{(r)}\delta_P^{1/2}$$

for some $\mu$ and $\Theta_H^{(r)}$ such that $\Theta_H^{(r)}$ is not generic. This aims to characterize non-generic unramified representations through induction from non-generic Theta representations, while explicitly excluding the special cases treated separately in the paper.