Rational Malcev conjecture for the pure cactus group

Let $\widehat{\Gamma_n}$ be the prounipotent completion of the pure cactus group $\Gamma_n$, and let $\psi_n^{\mathbb{Q}}:L_n\otimes\mathbb{Q}\to\operatorname{grLie}\widehat{\Gamma_n}$ be the induced surjective map. Rational Malcev conjecture. The map $\psi_n^{\mathbb{Q}}$ is an isomorphism, and $\Gamma_n$ injects into its prounipotent completion. These are the rational versions of the integral lower-central-series assertions and remain open.