Geometric comparison of complex and $p$-adic Arthur–Vogan packets

Let $n$, $r$, $k_i$, and $m_i$ be as above, and let $\psi^\BR$ denote the corresponding parameter on the complex side. On the $p$-adic side, let $\psi^p$ and the Arthur–Vogan packets $\Pi_{\psi^p}^{\text{A-V}}$ be as defined above. Geometric comparison conjecture. When $n$ is even, there is a bijection

$$\Pi_{\psi^\BR}^{\text{A-V}} \bijects \Pi_{\psi^p}^{\text{A-V}}.$$

When $n$ is odd, there is a bijection

$$\bigsqcup_{\substack{ p + q = n\\ p \equiv \frac{n-1}{2} \, {\rm mod} \, 2 }} \Pi_{\psi^\BR}^{\bU(p,q)} \bijects \Pi_{\psi^{p}}^{\text{A-V}}.$$

This proposes a direct comparison between the complex and $p$-adic Arthur–Vogan packets, with the parity of $n$ determining whether one complex packet or a disjoint union over unitary groups occurs. The source provides no resolution, so the conjecture remains open.