Gordon–Martino conjecture on unipotent families and Calogero–Moser fixed points

Let $W$ be the Weyl group of a reductive group with Frobenius action induced by $\tau$, let $k_{\mathrm{sp}}$ be the special parameter, and let $\mathcal Z_{k_{\mathrm{sp}}}$ be the corresponding Calogero–Moser space. Write $({\mathcal Z}_{k_{\mathrm{sp}}}^{\mathbb C^\times})^\tau$ for its $\tau$-fixed $\mathbb C^\times$-fixed points, and ${\mathrm{Fam}}_{\mathrm{un}}(W\tau)$ for the unipotent Lusztig families.

Unipotent-family conjecture. There exists a unique bijection

$$\Phi:({\mathcal Z}_{k_{\mathrm{sp}}}^{\mathbb C^\times})^\tau\xrightarrow{\sim}{\mathrm{Fam}}_{\mathrm{un}}(W\tau)$$

such that, for every $p$ in the source and every $\tau$-stable character $\chi\in(\mathfrak F_p^{k_{\mathrm{sp}}})^\tau$, the almost character $R_\chi$ belongs to the one-dimensional span $\mathbb C\Phi(p)$.

This is the principal proposed link between Calogero–Moser fixed-point geometry and unipotent representation theory. When $\tau=\operatorname{Id}_V$, the source states that it is equivalent to the Gordon–Martino conjecture; the general equivariant formulation remains open.