Jiang's conjecture on wavefront sets in local Arthur packets

Let $\mathrm G$ be a connected reductive group over $F$ and let $G=\mathrm G(F)$. Assume a local Arthur-packet theory for $G$. Let $\Psi^+(G)$ be the set of local Arthur parameters, let $\Pi_\psi$ be the packet attached to $\psi$, and let $\mathcal O_\psi$ be its geometric nilpotent orbit. Jiang's conjecture. For every $\psi\in\Psi^+(G)$, (i) for every $\pi\in\Pi_\psi$ and every $\mathcal O\in\overline{\mathfrak n}^{m}(\pi)$,

$$\mathcal O\leq d_{BV}(\mathcal O_\psi),$$

and (ii) if $G$ is quasi-split, at least one $\pi\in\Pi_\psi$ satisfies

$$\overline{\mathfrak n}^{m}(\pi)=\{d_{BV}(\mathcal O_\psi)\}.$$

The conjecture generalizes the Shahidi conjecture from tempered $L$-packets to arbitrary local Arthur packets. Its status depends on the existence and properties of local Arthur packets and is not established in general.