Twisted wave-front containment and orbit-dimension conjecture

Let $F$ be a non-Archimedean local field, let $\bf G$ be an algebraic reductive group over $F$, and let $G=\bf G(F)$. Let $\bf H\subset G$ be a spherical subgroup defined over $F$, with $H=\bf H(F)$, and let $\phi$ be a character of $H$. Assume that $H=S\ltimes N$, where $N$ is a parabolic nilradical. Write $\mathfrak s$ and $\mathfrak n$ for the Lie algebras of $S$ and $N$, and choose $\chi\in\mathfrak g^*$ such that $\chi|_{\mathfrak s}=0$ and $\chi|_{\mathfrak n}$ determines $\phi|_N$. For $\pi\in\operatorname{Irr}(G)$, let $\operatorname{WF}^{\max}(\pi)$ be the union of the maximal nilpotent orbits in the closure of its wave-front set, and let $\mathfrak h^{\bot}$ be the annihilator of the Lie algebra of $H$. Twisted wave-front conjecture. Under these assumptions, for every $\pi\in\operatorname{Irr}(G)_{(H,\phi)}$ one has

$$\operatorname{WF}^{\max}(\pi)\subset \mathbf{G}\cdot(\chi+\mathfrak h^{\bot}).$$

Moreover, for every coadjoint $\mathbf{G}$-orbit $\mathcal O$ intersecting $\operatorname{WF}^{\max}(\pi)$, one has

$$\dim\bigl(\mathcal O\cap(\chi+\mathfrak h^{\bot})\bigr)=\dim\mathcal O/2.$$

This is intended to extend the spherical wave-front prediction to representations with a character twist and to provide a Lagrangian-type dimension condition. The source gives no evidence resolving it.