Freeness conjecture for the Lie and enveloping algebras associated with the cactus group

Let $L_n$ be the quadratic Lie algebra generated by the elements $\mu_{ijk}$, and let $U_n=U(L_n)$ be its universal enveloping algebra. Freeness conjecture. The $\mathbb{Z}$-modules $U_n$ and $L_n$ are free. This conjecture is used in the paper to formulate stronger integral Koszulness and torsion conclusions; its general status is open.