Global coherence conjecture for wavefront sets of automorphic representations

Let $k$ be a number field and let $G$ be a reductive algebraic group defined over $k$. Let

$$\pi=\bigotimes_v\pi_v$$

be an automorphic representation, where $v$ ranges over the places of $k$, $k_v$ is the corresponding completion, and each $\pi_v$ is a smooth irreducible admissible representation of $G(k_v)$. For each place, let

$$\WF(\pi_v)\subset {\mathcal N}^*_{k_v}\subset {\mathfrak g}(k_v)^*$$

a closed $G(k_v)$-invariant cone. Choose the local characters $\psi_v$ so that

$$\prod_v\psi_v(x)=1,\qquad (x\in k).$$

Global coherence of wavefront sets. There should exist a coadjoint orbit depending on $\pi$,

$${\mathcal O}=G(k)\cdot\xi\subset {\mathfrak g}(k)^*,$$

with the property that

$$\WF(\pi_v)=\operatorname{Cone}_{k_v}({\mathcal O})$$

for every place $v$. This conjecture proposes that the local wavefront sets of an automorphic representation arise coherently by extending a single global coadjoint orbit to every completion of the number field.