Existence of the Plancherel-to-cell map for affine Hecke algebras

Let $\Pi$ range over the classes of representations appearing in the Plancherel formula, and let the two-sided cells be the cells of the affine Hecke algebra. A Plancherel-to-cell map is a map from these representation classes to the two-sided cells. Map conjecture. There exists a well-defined surjective map $\Omega$ from the set of classes of representations appearing in the Plancherel formula to the set of two-sided cells, generalising the map from Proposition~. This would extend the correspondence between Plancherel representations and two-sided cells beyond the case established by the cited proposition.