Conjecture on maximal Fourier coefficients of automorphic packets for symplectic groups

Let $\psi\in\widetilde{\Psi}_2(\operatorname{Sp}_{2n})$, and let $\widetilde{\Pi}_{\psi}(\epsilon_\psi)$ be the automorphic $L^2$-packet attached to $\psi$. Write $\underline{p}(\psi)$ for the partition attached to the global Arthur parameter, and let $\frak{p}^m(\pi)$ denote the set of partitions associated with Fourier coefficients of $\pi$. Let $\eta_{\mathfrak{g}^\vee,\mathfrak{g}}$ be the Barbasch–Vogan duality map from partitions for $\mathfrak{so}_{2n+1}(\mathbb{C})$ to partitions for $\mathfrak{sp}_{2n}(\mathbb{C})$. A symplectic partition is a partition of $2n$ satisfying the symplectic condition.

Conjecture 4.2. For every $\psi\in\widetilde{\Psi}_2(\operatorname{Sp}_{2n})$:

1. Any symplectic partition $\underline{p}$ of $2n$ with $\underline{p}>\eta_{\mathfrak{g}^\vee,\mathfrak{g}}(\underline{p}(\psi))$ 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 partition $\underline{p}\in\frak{p}^m(\pi)$ satisfies $\underline{p}\leq\eta_{\mathfrak{g}^\vee,\mathfrak{g}}(\underline{p}(\psi))$.
3. There exists at least one $\pi\in\widetilde{\Pi}_{\psi}(\epsilon_\psi)$ such that $\eta_{\mathfrak{g}^\vee,\mathfrak{g}}(\underline{p}(\psi))\in\frak{p}^m(\pi)$.

The conjecture predicts that the Barbasch–Vogan dual of the Arthur-parameter partition gives the precise maximal partition governing Fourier coefficients across the packet. Its status is not resolved by the supplied source context.