Okada's conjecture on canonical wavefront sets of unipotent representations

Let $G=\mathrm G(F)$, where $\mathrm G$ is a connected reductive algebraic group defined over $F$ and inner to a split group. Let ${}^{K}\mathrm{WF}(\pi)$ denote the unramified canonical wavefront set of a depth-zero representation, and let $d_S$ be Sommer duality on the corresponding orbit-component data. Okada's conjecture. For every unipotent representation $\pi$ of $G$, the set ${}^{K}\mathrm{WF}(\pi)$ is a singleton and

$$d_S({}^{K}\mathrm{WF}(\pi))=\mathcal O_{\phi_{\widehat\pi}},$$

where $\phi_{\widehat\pi}$ is the parameter of the Aubert–Zelevinsky dual. The conjecture predicts singleton wavefront behavior for unipotent representations with non-real infinitesimal parameter and identifies the singleton through Sommer duality. It is cited in the source as Okada's conjecture and is not resolved in general.