Conjecture on symplectic leaves of Calogero–Moser spaces for finite Coxeter groups

Let $W$ be a finite Coxeter group. The Calogero–Moser space $\cX_\bc(W)$ is stratified into finitely many symplectic leaves, naturally labeled by conjugacy classes $(W')$ of parabolic subgroups $W'$ of $W$. The geometric ordering on leaves is the ordering defined by containment of their closures, while the algebraic ordering is the ordering induced by inclusions of parabolic subgroups.

Symplectic-leaf conjecture. Each conjugacy class of parabolic subgroups $(W')$ labels at most one symplectic leaf, and the geometric ordering on leaves equals the algebraic ordering.

The paper establishes these descriptions for types $A$, $B$, $D$ and $I_2(m)$. The assertion is proposed as a general conjecture for finite Coxeter groups, including the exceptional types.