Arthur-parameter bound for maximal Fourier coefficients

Let $G\in\widetilde{{\mathcal {E}}}_{\mathrm{sim}}(N)$ be an $F$-quasisplit classical group. For a global Arthur parameter $\psi\in\widetilde{\Psi}_2(G)$, write

$$\psi=(\tau_1,b_1)\boxplus\cdots\boxplus(\tau_r,b_r),$$

where $\tau_i\in{\mathcal {A}}_{\mathrm{cusp}}({\mathrm{GL}}_{a_i})$, and let $\underline{p}(\psi)=[(b_1)^{(a_1)}\cdots(b_r)^{(a_r)}]$. Let $\eta_{{\frak{g}^\vee,\frak{g}}}$ be the Barbasch–Vogan duality map, and let $\widetilde{\Pi}_{\psi}(\epsilon_\psi)$ be the automorphic $L^2$-packet attached to $\psi$.

Arthur-parameter conjecture. The following hold: (1) if $\underline{p}>\eta_{{\frak{g}^\vee,\frak{g}}}(\underline{p}(\psi))$, then $\underline{p}$ does not belong to $\frak{p}^m(\pi)$ for any $\pi\in\widetilde{\Pi}_{\psi}(\epsilon_\psi)$; (2) for every $\pi\in\widetilde{\Pi}_{\psi}(\epsilon_\psi)$, every $\underline{p}\in\frak{p}^m(\pi)$ satisfies $\underline{p}\leq\eta_{{\frak{g}^\vee,\frak{g}}}(\underline{p}(\psi))$; and (3) at least one member $\pi$ of the packet satisfies $\eta_{{\frak{g}^\vee,\frak{g}}}(\underline{p}(\psi))\in\frak{p}^m(\pi)$.

This conjecture predicts that the Barbasch–Vogan dual of the partition determined by the Arthur parameter is the precise maximal bound, and is attained by at least one packet member. The source gives no resolution status.