Geometric wave-front set singleton conjecture

Let $F$ be a nonarchimedean local field, let $G$ be a reductive algebraic group over $F$, and let $\bar{F}$ be an algebraic closure of $F$. For an irreducible admissible representation $\pi$ of $G(F)$, let $\overline{\operatorname{WF}}^{rat}(\pi)$ denote the set of $\operatorname{Ad}(G(\bar{F}))$-orbits in $\mathfrak{g}(\bar{F})$ that meet the rational wave-front set $\operatorname{WF}^{rat}(\pi)$. Geometric wave-front set singleton conjecture. For any irreducible admissible representation $\pi$ of $G(F)$, the set $\overline{\operatorname{WF}}^{rat}(\pi)$ is a singleton. This long-standing conjecture was proposed and proved for $\mathrm{GL}_n$; it is refuted in general by the counterexample discussed in the paper.