The parabolic symmetrising trace conjecture for complex reflection groups

Let $W$ be a complex reflection group, let $I$ be an intersection of reflecting hyperplanes of $W$, and let $W_I$ be the corresponding parabolic subgroup. Let $d4d5(W)$ be the generic Hecke algebra of $W$, let $d4d5_I(W)$ be its parabolic Hecke subalgebra, and let $d70f$ be the canonical symmetrising trace on $d4d5(W)$. The parabolic symmetrising trace conjecture. The restriction

$$d70f|_{d4d5_I(W)}$$

is the canonical symmetrising trace on $d4d5_I(W)$.

This conjecture is the parabolic counterpart of the Broué–Malle–Michel symmetrising trace conjecture and is fundamental for induction and restriction arguments. The source proves it for the listed exceptional groups, but does not establish it for all complex reflection groups.