Generic Summand Conjecture for Bessel modules

Let $G_n$ be a group in the Bessel descent setting, let ${\mathcal {A}}_{\mathrm{cusp}}(G_n)$ be its cuspidal automorphic representations, and let $\widetilde{\Phi}_2(G_n^*)$ denote the relevant generic global Arthur parameters. For $\pi\in{\mathcal {A}}_{\mathrm{cusp}}(G_n)$ with a $G_n$-relevant generic global Arthur parameter $\phi\in\widetilde{\Phi}_2(G_n^*)$, a cuspidal realization ${\mathcal {C}}_\pi$ is a realization of $\pi$ in $L^2_{\mathrm{cusp}}(G_n)$. Its first occurrence index is denoted by $\ell_0({\mathcal {C}}_\pi)$; ${\mathcal {O}}_{\ell_0}$ is a $k$-rational orbit associated to the corresponding partition, and ${\mathcal {F}}^{\psi_{{\mathcal {O}}_{\ell_0}}}$ is the associated Bessel module.

Generic Summand Conjecture. There exists a cuspidal realization ${\mathcal {C}}_\pi$ with first occurrence index $\ell_0$ and a suitable $k$-rational orbit ${\mathcal {O}}_{\ell_0}$ such that there are $\sigma\in{\mathcal {A}}_{\mathrm{cusp}}(H_{\ell_0^-}^{{\mathcal {O}}_{\ell_0}})$ with a relevant generic global Arthur parameter, and a cuspidal realization ${\mathcal {C}}_\sigma$ of $\sigma$ in $L^2_{\mathrm{cusp}}(H_{\ell_0^-}^{{\mathcal {O}}_{\ell_0}})$, for which

$$\left\langle {\mathcal {F}}^{\psi_{{\mathcal {O}}_{\ell_0}}}(\varphi_\pi),\varphi_\sigma\right\rangle_{H_{\ell_0^-}^{{\mathcal {O}}_{\ell_0}}}\neq 0$$

for suitable $\varphi_\pi\in{\mathcal {C}}_\pi$ and $\varphi_\sigma\in{\mathcal {C}}_\sigma$.

This is presented as a preliminary form of the spectrum problem for the first Bessel module and is attributed in the source to an earlier conjecture. The source gives no resolution status.