The BMM symmetrising trace conjecture for complex reflection groups

Let $W$ be a complex reflection group with generic Hecke algebra $\mathcal{H}(W)$ over $\mathbb{Z}[\mathbf{u},\mathbf{u}^{-1}]$. Let $B(W)$ be its braid group, let $\boldsymbol{\pi}$ be the distinguished braid-group element used in the conjecture, and let $\beta\mapsto T_\beta$ be the natural surjection $B(W)\to\mathcal{H}(W)$. Let $x\mapsto x^*$ be the automorphism of $\mathbb{Z}[\mathbf{u},\mathbf{u}^{-1}]$ induced by $\mathbf{u}\mapsto\mathbf{u}^{-1}$. The BMM symmetrising trace conjecture. There exists a canonical symmetrising trace $\tau$ on $\mathcal{H}(W)$ such that: (1) $\tau$ specialises to the canonical symmetrising trace on the group algebra of $W$ under $u_{\mathcal{C},j}\mapsto\zeta_{e_{\mathcal{C}}}^{j}$; and (2) for every $\beta\in B(W)$,

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

The conjecture supplies a canonical trace with both the required group-algebra specialisation and duality condition. It is known for the infinite series $G(l,p,n)$ only that a trace satisfying the first condition exists, while the full conjecture has been proved for $G_4$, $G_{12}$, $G_{22}$ and $G_{24}$, and this paper proves it for $G_4,\ldots,G_8$.