Fixed-point conjecture for Calogero-Moser spaces

Let $W$ act as a complex reflection group on $V$, let $\tau$ be a finite-order element of the normalizer of $W$ in $\operatorname{GL}_{\mathbb C}(V)$, and let ${\boldsymbol{\mathcal Z}}^\tau$ be the reduced fixed-point variety. A reflection subquotient $(V',W')$ consists of a subspace $V'\subseteq V$ and a quotient $W'=N'/N'_1$ of a subgroup $N'$ of the stabilizer of $V'$ in $W$, where $N'_1$ is the kernel of the action on $V'$ and $W'$ acts as a reflection group. Fixed-point conjecture. For every irreducible component ${\boldsymbol{\mathcal X}}$ of ${\boldsymbol{\mathcal Z}}^\tau$, there exist a reflection subquotient $(V',W')$ and a linear map $\varphi:{\boldsymbol{\mathcal C}}^\tau\rightarrow{\boldsymbol{\mathcal C}}(V',W')$ such that

$${\boldsymbol{\mathcal X}}\simeq{\boldsymbol{\mathcal Z}}(V',W')\times_{{\boldsymbol{\mathcal C}}(W')}{\boldsymbol{\mathcal C}}^\tau.$$

This conjecture predicts that fixed-point components under finite-order normalizer automorphisms are themselves parameterized families of Calogero-Moser spaces for reflection subquotients; the source does not report a general proof.