Inductive kernel conjecture for the Lie and enveloping algebras of the cactus group

Let $U_n=U(L_n)$, where $L_n$ is the quadratic Lie algebra generated by the elements $\mu_{ijk}$. Consider the natural morphisms $U_{n+1}\to U_n$ and $L_{n+1}\to L_n$ sending $\mu_{ijn}$ to zero and fixing $\mu_{ijk}$ for $i,j,k<n$. Inductive kernel conjecture. The kernel of $U_{n+1}\to U_n$ is a free $U_{n+1}$-module, and the kernel of $L_{n+1}\to L_n$ is a free Lie algebra with infinitely many generators. These assertions are motivated by computations of Hilbert-series and character quotients; they remain open.