Gordon–Martino conjecture on Calogero–Moser and Kazhdan–Lusztig cells

Let $W$ be a Weyl group, let ${\bf c}$ be the parameter, and let ${\bf L}:W\to\mathbb{Q}$ be the weight function determined by ${\bf L}(s)=c_s$ for each simple reflection $s\in\mathcal{S}$. Let $CM_{\bf c}$ be the partition of $\operatorname{Irr}W$ into fibres of the map to the nilpotent points of the Calogero–Moser space, and let $KL_{\bf L}$ be the partition into two-sided Kazhdan–Lusztig cells. For a $KL_{\bf L}$-cell $\mathcal{F}$, write $M_{\mathcal{F}}$ for the corresponding closed point of the fibre $\Upsilon_{\bf c}^{*}(0)$.

Gordon–Martino conjecture. There is a natural identification of the $CM_{\bf c}$-partition and the $KL_{\bf L}$-partition, induced by attaching a $KL_{\bf L}$-cell to an irreducible $W$-representation via the asymptotic algebra $J$. Moreover,

$$\dim_{\mathbb{C}}\bigl(\Upsilon_{\bf c}^{*}(0)_{M_{\mathcal{F}}}\bigr)=|\mathcal{F}|.$$

This conjecture predicts both the coincidence of Calogero–Moser blocks with unequal-parameter two-sided cells and the length of each corresponding local fibre. The source provides the definitions and attribution to Lusztig but no resolution evidence.