Broué–Malle–Michel's symmetrising trace conjecture for generic Hecke algebras

Let $W$ be a complex reflection group, let $R(W)$ be its ring of definition, and let $\mathcal{H}(W)$ be its generic Hecke algebra. Let $B(W)$ be the braid group, let $\boldsymbol{\pi}$ be the full-twist element, and let $T_b$ denote the image of $b\in B(W)$ in $\mathcal{H}(W)$. Broué–Malle–Michel's symmetrising trace conjecture. There exists a linear map

$$\tau:\mathcal{H}(W)\rightarrow R(W)$$

satisfying the three conditions stated in the source: it is a symmetrising trace; it specialises to the canonical trace on the group algebra; and

$$\tau(T_{b^{-1}})^*=\frac{\tau(T_{b\boldsymbol{\pi}})}{\tau(T_{\boldsymbol{\pi}})}$$

for all $b\in B(W)$, with the involution $u_{\mathcal{C},j}\mapsto u_{\mathcal{C},j}^{-1}$.

The source reports that this conjecture remains open for most complex reflection groups, although it has been proved for several exceptional groups and partially established for the infinite series.