Calogero–Moser fixed-point and unipotent cuspidality conjecture

Assume the unipotent-family bijection $\Phi$ exists. Let $p\in({\mathcal Z}_{k_{\mathrm{sp}}}^{\mathbb C^\times})^\tau$ and suppose that some $d$-cuspidal unipotent representation $\lambda\in\Phi(p)\subset{\mathrm{Unip}}(W\tau)$ lies in the family attached to $p$. Let $\tau_d=\zeta_d\tau$ denote the corresponding finite-order twist, and call a point $\tau_d$-cuspidal when it is a zero-dimensional symplectic leaf.

Cuspidality conjecture. Then $p$ is $\tau_d$-cuspidal.

This predicts that $d$-cuspidality on the finite reductive-group side is detected by cuspidality of Calogero–Moser fixed points. The source explicitly notes that the converse fails in general, even for $d=1$, while the forward implication remains conjectural.