The Bessel-period obstruction to first occurrence

Let $G_n$ and $H_m$ be the groups in the Bessel descent setting, and let ${\mathcal {A}}_{\mathrm{cusp}}(G_n)$ and ${\mathcal {A}}_{\mathrm{cusp}}(H_m)$ denote their cuspidal automorphic representations. Assume that $\pi\in{\mathcal {A}}_{\mathrm{cusp}}(G_n)$ has a generic global Arthur parameter $\phi_n$ as in the source, and that for $\sigma\in{\mathcal {A}}_{\mathrm{cusp}}(H_m)$ the Bessel period ${\mathcal {B}}^{{\mathcal {O}}_{\ell}}(\varphi_\pi,\varphi_\sigma)$ is nonzero for suitable cusp forms. Suppose the global Arthur parameter $\psi_m$ of $\sigma$ contains $(\xi_1,2)\boxplus\cdots\boxplus(\xi_k,2)$ with $k>0$, and let ${\mathfrak {r}}_n={\mathfrak {r}}(G_n)$.

Bessel-period obstruction conjecture. The integer $\ell$, with $0\leq\ell\leq {\mathfrak {r}}_n$, is not the first occurrence index of $\pi$.

This predicts that a nonzero Bessel period against a parameter containing a nontrivial collection of summands of the form $(\xi_i,2)$ cannot occur at the first occurrence level. The source provides no resolution status.