The hyperelliptic orbital-integral conjecture for classical groups

Orbital integrals for classical groups are expected to be related only to hyperelliptic curves. The precise formulation below concerns the cohomology of Hessenberg varieties. Hyperelliptic cohomology conjecture. For $\hat{\mathcal{H}}_{e}(\gamma)$ as in this section, with $G_{\bar{k}}$ having only isogenous factors of types $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$, $\mathrm{D}$, and $\mathrm{G}_2$, the semisimplification of the $\overline{\mathbb{Q}}_{\ell}[\operatorname{Gal}(\bar{k}/k)]$-module

$$H^*_c(\hat{\mathcal{H}}_{e}(\gamma)\otimes\bar{k},\overline{\mathbb{Q}}_{\ell})$$

is isomorphic to a direct sum of some submodules of certain tensor products of $H^1_c(-,\overline{\mathbb{Q}}_{\ell})$ for hyperelliptic curves and $H^0_c(-,\overline{\mathbb{Q}}_{\ell})$ for finite étale schemes. This would give a precise cohomological form of the expectation that orbital integrals for classical groups involve only hyperelliptic curves; the source gives no resolution, while examples in the surrounding discussion indicate that non-hyperelliptic curves occur in type $\mathrm{E}$ outside the stated types.