Linearity and group-theoretic properties of braid and pure braid groups of complex reflection groups

Let $W$ be an irreducible complex reflection group, and let $B$ and $P$ denote respectively its braid group and pure braid group. Let the properties enumerated in the referenced theorem be the stated linearity and group-theoretic properties of these groups.

Conjecture. The groups $B$ and $P$ are linear and satisfy all the properties enumerated in that theorem whenever $W$ is an irreducible complex reflection group.

The claim extends the corresponding result beyond irreducible Coxeter groups of type ADE. Its status is not determined by the supplied text.