The freeness conjecture for Hecke algebras of complex reflection groups

Let $W$ be a finite complex reflection group, let $W'$ be a parabolic subgroup of $W$, and let ${\mathcal H}(W)$ and ${\mathcal H}(W')$ be their Hecke algebras over $A=\mathbb{Z}[x,x^{-1}]$. Freeness conjecture. The algebra ${\mathcal H}(W)$ should be a free ${\mathcal H}(W')$-module of rank $[W:W']$. This is an analogue of the expected freeness properties of ordinary Iwahori–Hecke algebras; the source gives no resolution of the conjecture in this generality.