Bonnafé's fixed-point conjecture for two-sided cells

Let $G$ be a complex reductive group with Cartan subalgebra $\mathfrak{t}$, Weyl group $W$, and $A=\mathbb{C}[\mathfrak{t}\oplus\mathfrak{t}^*]^{\mathrm{sgn}}$. Set $\Delta=\prod_{\alpha\in\Phi^+}\alpha$ and define

$$X_{\mathfrak{g},\mathrm{sgn}}:=\operatorname{Proj}\bigoplus_{d\geq 0}(\Delta A)^d.$$

The construction has a natural $\mathbb{C}^\times$-action. Bonnafé's fixed-point conjecture. There is a bijection

$$\{\text{two-sided cells in }W\}\leftrightarrow X_{\mathfrak{g},\mathrm{sgn}}^{\mathbb{C}^\times}.$$

Consequently, there are only finitely many fixed points, depending only on $W$; the claim is known for $\mathfrak{g}=\mathfrak{gl}_n$ and for type $B_2$, but remains open in general.