The residual torsion-free nilpotence conjecture for pure braid groups of reflection arrangements

Let $W\subset \mathrm{GL}_n(\mathbb{C})$ be a finite pseudo-reflection group, and let $\mathcal{A}$ be the corresponding hyperplane arrangement in $\mathbb{C}^n$. Consider the fundamental group of the arrangement complement.

Residual torsion-free nilpotence conjecture. The fundamental group of the complement

$$\mathbb{C}^n\setminus\bigcup \mathcal{A}$$

is residually torsion-free nilpotent.

For the arrangement complement, this fundamental group is the pure braid group associated with $W$. The claim would extend residual torsion-free nilpotence beyond the cases discussed as known in the source, and its resolution is not given.