The symmetrising-form conjecture for generic Hecke algebras

Let $W$ be a finite complex reflection group, let $\mathcal H(W,\mathbf u)$ be its generic Hecke algebra over the base ring $A$, and let $f_\chi$ be the Schur element associated with each irreducible character $\chi$. Define the canonical trace form by

$$t_{W,\mathbf u}(h):=\sum_{\chi\in\operatorname{Irr}(\mathcal H(W,\mathbf u))}\frac{1}{f_\chi}\chi(h).$$

Symmetrising-form conjecture. The form $t_{W,\mathbf u}$ takes values in $A$ and is a symmetrising form on $\mathcal H(W,\mathbf u)$. This is the Hecke-algebra conjecture attributed in the source to Brou'e--Malle--Michel; the paper treats it as a conjecture and notes that it is known in many cases.