Wavefront set conjecture for constituents of regular unramified principal series

Let $\tilde{G}$ be a persistent $n$-fold cover, and let $\nu\in X\otimes\mathbf{R}$ be regular with $\Phi(\nu)\subset\Delta$. For $S\subset\Phi(\nu)$, let $W(S)$ be the Weyl subgroup generated by $S$, let $W_{\tilde{\nu}}^S$ be the integral Weyl subgroup associated with a saturation $\tilde{\nu}$ of $\nu$, and let $\varepsilon_{\tilde{\nu}}^S$ be its sign character. Let $I(\pi^\dag,\nu)$ be the regular unramified principal series and let $\pi_S$ be a constituent. Wavefront set conjecture. For every constituent $\pi_S$ of $I(\pi^\dag,\nu)$ with $S\subset\Phi(\nu)$, one has

$$\mathcal{N}_{\rm tr}^{\rm \max}(\pi_S)\otimes\overline{F}=\left\{\mathcal{O}_{\rm Spr}(j_{W_{\tilde{\nu}}^S}^W(\varepsilon_{\tilde{\nu}}^S))\right\}.$$

This generalizes the corresponding assertion for the theta representation; the supplied text gives no resolution status, although the linear-group case is stated to be proved by Mœglin and Waldspurger.