BMM symmetrising trace conjecture for generic Hecke algebras

Let $W$ be an irreducible complex reflection group, let $B(W)$ be its braid group, let $R=\mathbb{Z}[\mathbf{u},\mathbf{u}^{-1}]$, and let $\mathcal{H}(W)$ be its generic Hecke algebra. Let $\boldsymbol{\pi}$ be the distinguished central element defined from the centre of $W$. A symmetrising trace is an $R$-linear trace $\tau:\mathcal{H}(W)\to R$ inducing the associated symmetrising form. For $\beta\in B(W)$, write $T_\beta$ for its image in $\mathcal{H}(W)$, and let $x\mapsto x^*$ be the automorphism of $R$ given by $\mathbf{u}\mapsto\mathbf{u}^{-1}$.

BMM symmetrising trace conjecture. There exists a symmetrising trace $\tau:\mathcal{H}(W)\to R$ such that:

1. $\tau$ specialises to the canonical symmetrising trace on the group algebra of $W$ when $u_{\mathcal{C},j}\mapsto\eta_{e_\mathcal{C}}^j$;
2. for every $\beta\in B(W)$,

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

The conjecture concerns a canonical trace structure on generic Hecke algebras of complex reflection groups and is attributed in the paper to Broué, Malle and Michel. Its resolution status is not stated in the supplied material.