Malle–Michel's lifting conjecture for Hecke algebra bases

Let $W$ be a complex reflection group, let $B(W)$ be its braid group, let $\mathcal{H}(W)$ be its generic Hecke algebra, and let $\tau$ be a canonical symmetrising trace on $\mathcal{H}(W)$. Malle–Michel's lifting conjecture. There exists a section

$$W\rightarrow\boldsymbol{W}\subset B(W),\qquad w\mapsto\boldsymbol{w},$$

of the quotient map $B(W)\rightarrow W$ such that $1\in\boldsymbol{W}$ and

$$\tau(T_{\boldsymbol{w}})=\delta_{1\boldsymbol{w}}$$

for every $\boldsymbol{w}\in\boldsymbol{W}$.

This conjecture seeks a lifted copy of the reflection group whose braid lifts have the expected trace values. The supplied text attributes it to Malle and Michel but does not state a general resolution.