The centre-surjectivity conjecture for Hecke algebras of complex reflection groups

Let $A=\mathbb{Z}[x,x^{-1}]$, let $R$ be a commutative local $A$-algebra, and let $F$ be its residue field. Let ${\mathcal H}$ be the Hecke algebra of a finite complex reflection group. Centre-surjectivity conjecture. The canonical morphism

$$Z(R{\mathcal H})\longrightarrow Z(F{\mathcal H})$$

is surjective. This property controls the behaviour of the centre under scalar extension and is equivalent to the corresponding property for class functions on a symmetric algebra. The source states that the conjecture is true when $W$ is a Weyl group, while the general case remains open.