Supercuspidal wavefront-set conjecture

Let $\pi(\lambda,s,{{\mathcal O}}^\vee_{\lambda,s},{{\mathcal L}}^\vee)$ be an irreducible supercuspidal ${\mathbf G}^\omega({\mathsf k})$-representation in the $L$-packet $\Pi(\varphi)=\Pi(\lambda,s,{{\mathcal O}}^\vee_{\lambda,s})$, and set

$${\mathbb O}^\vee_{\varphi}=G^\vee\cdot{{\mathcal O}}^\vee_{\lambda,s}.$$

Supercuspidal wavefront-set conjecture. One has

$${}^{\bar {\mathsf k}}\mathsf{WF}(\pi)\leq d^\vee({\mathbb O}^\vee_{\varphi}),$$

or equivalently ${\mathbb O}^\vee_{\varphi}\leq d({}^{\bar {\mathsf k}}\mathsf{WF}(\pi))$. If ${\mathbf G}^\omega({\mathsf k})$ is split and $\pi$ is depth-zero compactly induced from a cuspidal representation of the maximal hyperspecial parahoric subgroup, then

$${}^{\bar {\mathsf k}}\mathsf{WF}(\pi)=\{d^\vee({\mathbb O}^\vee_{\varphi})\}.$$

The first assertion is a particular case of the paper's main conjecture. The equality is verified for depth-zero supercuspidal representations of symplectic, special orthogonal, and unitary groups, while the general statement remains open.