Wavefront-set upper-bound conjecture for Langlands parameters

Let ${\mathbf G}^\omega({\mathsf k})$ be a reductive $p$-adic group, let $\pi(\lambda,s,{{\mathcal O}}^\vee,{{\mathcal L}}^\vee)$ be an irreducible representation in the $L$-packet $\Pi(\varphi)=\Pi(\lambda,s,{{\mathcal O}}^\vee)$, and let ${}^{\bar {\mathsf k}}\mathsf{WF}(\pi)$ denote its geometric wavefront set. For sets of nilpotent orbits, write $\mathcal S_1\leq\mathcal S_2$ when every orbit in $\mathcal S_1$ is contained in the closure of some orbit in $\mathcal S_2$. Let $d$ and $d^\vee$ be the relevant duality maps, and let $G^\vee\cdot{{\mathcal O}}^\vee$ denote the $G^\vee$-saturation of the dual nilpotent orbit. Wavefront-set upper-bound conjecture. For every such $\pi$,

$${}^{\bar {\mathsf k}}\mathsf{WF}({\mathsf{AZ}}(\pi))\leq d^\vee(G^\vee\cdot{{\mathcal O}}^\vee).$$

Equivalently,

$${{\mathcal O}}^\vee\subset\overline{d({}^{\bar {\mathsf k}}\mathsf{WF}({\mathsf{AZ}}(\pi)))}.$$

The conjecture relates geometric wavefront sets of Aubert–Zelevinsky duals to nilpotent orbits in Langlands parameters. It generalizes known results for several classes of unipotent and supercuspidal representations, but the general assertion remains open.