Injectivity conjecture for the map from the cactus Lie algebra to the braid Lie algebra

Let $L_n$ be the quadratic Lie algebra associated with the pure cactus group, let $\mathbf{L}_n$ be the corresponding braid Lie algebra, and let $\xi_n:L_n\to\mathbf{L}_n$ be defined by $\xi_n(\mu_{ijk})=[t_{ij},t_{jk}]$. Injectivity conjecture. The map $\xi_n$ is injective. The paper proves that its rationalization factors through the graded Lie algebra associated with the prounipotent completion, but injectivity remains open.