Triviality of the higher comparison kernels P_n

Prove that P_n(G) is trivial for every n whenever G is a free group, where P_n(G) is the kernel of the natural surjective homomorphism from the simple-bracket subgroup \hat{T}_n(G) of the free semi-multiplicative Lie algebra to the corresponding subgroup T_n(G) of the free multiplicative Lie algebra.

Background

The paper distinguishes the free semi-multiplicative Lie algebra from the free multiplicative Lie algebra. Their simple-bracket subgroups \hat{T}_n(G) and T_n(G) are related by a natural surjection, whose kernel P_n(G) defines a functor from groups to groups.

The kernel is known to be trivial for n=2 in all groups and, under additional hypotheses, for n=3. The authors explicitly conjecture triviality for every n when G is free, which would indicate that imposing the fourth multiplicative Lie identity introduces no further discrepancy in these higher simple-bracket components for free groups.