Expectation for the stable wavefront set of a theta representation

Let $F$ be a $p$-adic field with $p\nmid n$, let $\overline{G}$ be a persistent $n$-fold covering group, let $\nu\in X\otimes\mathbb{R}$ be exceptional, and let $\widetilde{\nu}\in X\otimes\mathbb{R}$ be a saturation of $\nu$. Let $\sigma^{\mathscr{X}}:W\to {\rm Perm}(\mathscr{X}_{Q,n})$ be the permutation representation given by the twisted Weyl action. For the Harish-Chandra local character expansion of $\Theta(\pi^\dagger,\nu)$, write $\mathcal{N}_{\rm tr}^{\rm \max}(\Theta(\pi^\dagger,\nu))$ for the maximal orbits occurring in the expansion.

Stable wavefront-set conjecture. One has

$$\mathcal{N}_{\rm tr}^{\rm \max}(\Theta(\pi^\dagger,\nu))\otimes F^{\rm al}=\left\{\mathcal{O}_{\rm Spr}^{\mathbf{G}}\bigl(j_{W_{\widetilde{\nu}}}^{W}(\varepsilon_{\widetilde{\nu}})\bigr)\right\}$$

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

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

This predicts the stable wavefront set and the corresponding character-expansion coefficient for theta representations at exceptional parameters. The supplied text describes it as an expectation and gives no evidence that it has been proved or disproved.