The upper bound conjecture for wavefront sets of representations

Let $H$ be a split classical group over $F$, let $u$ denote the nilpotent orbit of the image of

$$\begin{pmatrix}0&1\\0&0\end{pmatrix}$$

under the derivative of the relevant $\mathrm{SL}_2(\mathbb{C})$-factor, and let $d_{H^\vee}(\operatorname{Ad}(H^\vee)\nu)$ denote the corresponding dual nilpotent orbit. For $\phi\in\Phi(H(F))$, write $\widehat\phi$ for the parameter obtained from the second $\mathrm{SL}_2(\mathbb{C})$-factor and $N_{\widehat\phi}$ for its associated nilpotent element. The upper bound conjecture. For any $\pi\in\Pi_\phi$, if $\mathfrak{o}_H^\mathrm{st}\in\bar{\mathcal{N}}(\hat\pi)$, then

$$\mathfrak{o}_H^\mathrm{st}\leq d_{H^\vee}(\operatorname{Ad}(H^\vee)N_{\widehat\phi}).$$

Moreover, if $\phi$ is tempered, then

$$\bar{\mathcal{N}}(\hat\pi)^{\max}=\{d_{H^\vee}(\operatorname{Ad}(H^\vee)N_{\widehat\phi})\}$$

for some $\pi\in\Pi_\phi$. This conjecture concerns the wavefront upper bound predicted by the local Langlands correspondence. It is known for general linear groups, depth-zero simple supercuspidal representations of classical groups, certain unipotent representations with real infinitesimal character, and the exceptional group $G_2$; the general case remains open.