Integral lower-central-series conjecture for the pure cactus group

Let $\Gamma_n$ be the pure cactus group, let $\Gamma_n^k$ denote its lower central series, and let $\mathcal{L}_n$ be the associated graded Lie algebra after quotienting by $2$-torsion. Let $L_n$ be the quadratic Lie algebra and let $\psi_n:L_n\to\mathcal{L}_n$ be the natural surjective homomorphism. Integral lower-central-series conjecture. The map $\psi_n$ is an isomorphism; assuming the freeness conjecture, $\mathcal{L}_n$ is a free $\mathbb{Z}$-module and the only torsion in the lower central series of $\Gamma_n$ is $2$-torsion. Moreover,

$$\bigcap_{k\geq 1}\Gamma_n^k=\{1\},$$

so $\Gamma_n$ is residually nilpotent. The paper establishes surjectivity of $\psi_n$, but injectivity and residual nilpotence remain open.