Wavefront set conjecture for theta representations

Let $F$ be $p$-adic with $p mid n$. Let $\tilde{G}$ be a persistent $n$-fold covering group. Let $u\in X\otimes \mathbf{R}$ be exceptional and let $\tilde{ u}\in X\otimes \mathbf{R}$ be a saturation of $u$. Let $\Theta(\pi^\dag,\nu)$ be the \theta representation and let $\mathcal{N}_{\rm tr}^{\rm max}(\Theta(\pi^\dag,\nu))$ denote its maximal trace-norm nilpotent orbits; let $\mathcal{O}_{\rm Spr}$ be the nilpotent orbit attached by the Springer correspondence, $j_{W_{\tilde{\nu}}}^W$ the Macdonald $j$-induction, $\varepsilon_{\tilde{\nu}}$ the sign character, and $\sigma^{\mathscr{X}}$ the twisted Weyl permutation representation. Wavefront set conjecture. In the Harish-Chandra local character expansion of $\Theta(\pi^\dag,\nu)$, one has

$$\mathcal{N}_{\rm tr}^{\rm \max}(\Theta(\pi^\dag,\nu))\otimes \overline{F}=\left\{\mathcal{O}_{\rm Spr}(j_{W_{\tilde{\nu}}}^W(\varepsilon_{\tilde{\nu}}))\right\}$$

and, for every orbit $\mathcal{O}\in\mathcal{N}_{\rm tr}^{\rm \max}(\Theta(\pi^\dag,\nu))$,

$$c_{\mathcal{O}}=\left\langle j_{W_{\tilde{\nu}}}^W(\varepsilon_{\tilde{\nu}}),\varepsilon_W\otimes\sigma^{\mathscr{X}}\right\rangle_W.$$

This predicts the maximal nilpotent orbit and its coefficient in the local character expansion of theta representations; the supplied text gives no resolution status.