Calogero–Moser and Harish–Chandra symplectic-leaf correspondence conjecture

Assume the unipotent-family and cuspidality conjectures for $W$ and all its parabolic subgroups. Fix a $d$-cuspidal pair $(P,\lambda)$ and let $p$ be the Calogero–Moser point corresponding to the Lusztig family of $\lambda$. Let $\mathcal S_{P,p}$ be the associated symplectic leaf, and assume the relevant normalizer with $\lambda$ equals the full normalizer.

Harish–Chandra compatibility conjecture. The maps from irreducible characters of the normalizer to Calogero–Moser fixed points, from those characters to unipotent representations via $d$-Harish–Chandra theory, and from unipotent representations to fixed points via $\Phi^*$ form a commutative diagram:

$$\Phi^*\circ {\mathrm{H}}{\mathrm{C}}^{W,P,\lambda}=\psi_{\mathrm{fix}}\circ\mathfrak z_{k_{P,p}}.$$

This asserts compatibility between local Harish–Chandra parametrization and the global Calogero–Moser/unipotent correspondence. It is conditional on the preceding conjectures and remains open.