The pure braid group criterion for hyperplane arrangements

Let $K$ be the subgroup of $\uppi_{1}(\mathbb{C}^{n}\backslash \mathcal{H}_{\mathbb{C}})$ generated by the $J$-twists, where $J$ ranges over all subsets of the flopping curves, and let $\mathcal{H}$ be the associated hyperplane arrangement. The pure braid group criterion.

$$K=\uppi_{1}(\mathbb{C}^{n}\backslash \mathcal{H}_{\mathbb{C}})\quad\Longleftrightarrow\quad\mathcal{H}\text{ is a root system of a semisimple Lie algebra}.$$

The claim would identify precisely when the twists generate the full fundamental group and would provide new generating sets in cases where the arrangement comes from a semisimple Lie algebra. The paper indicates that verifying the equality in general is group-theoretically difficult.