Nilpotent-orbit filtration conjecture for affine Hecke algebras

Let $G$ be a connected reductive algebraic group over ${\mathbb{C}}$ with simply-connected derived group, let $\tilde{W}_a$ be its extended affine Weyl group, and let $H(\tilde{W}_a)$ be the associated Hecke algebra. Let $Z$ be Steinberg's triple variety, let ${\mathcal N}$ be the nilpotent variety of the Lie algebra of $G$, and let $\Phi:H(\tilde{W}_a)\to K^{G\times{\mathbb{C}}^*}(Z)$ be the geometric realization. For each nilpotent orbit $O$, let ${\mathcal C}_O$ be the corresponding two-sided cell of $\tilde{W}_a$, and write $H(\tilde{W}_a)_{\underset{LR}{\leq}{\mathcal C}_O}$ for the corresponding two-sided-cell filtration piece. Nilpotent-orbit filtration conjecture. For every nilpotent orbit $O$, one has

$$\Phi\bigl(H(\tilde{W}_a)_{\underset{LR}{\leq}{\mathcal C}_O}\bigr)=K^{G\times{\mathbb{C}}^*}(Z_{\overline{O}}).$$

This is a stronger geometric compatibility statement relating the two-sided-cell filtration of the affine Hecke algebra to the filtration by closures of nilpotent orbits. It is introduced as a natural expectation, with the source citing earlier work for related results; no resolution of the full assertion is supplied here.