Isomorphism of the free multiplicative Lie algebra commutator maps

Prove that, for every free group G and every positive integer n, the restriction homomorphism η_n from the subgroup T_n(G) of the free multiplicative Lie algebra LF(G) onto the lower-central-series subgroup γ_n(G) is an isomorphism.

Background

For a group G, the free multiplicative Lie algebra LF(G) admits a natural surjective homomorphism η_n from its subgroup P_n(G), or equivalently the corresponding simple-bracket subgroup T_n(G), onto γ_n(G). This map sends iterated multiplicative Lie products to iterated group commutators.

The paper identifies the assertion that η_n is an isomorphism for free groups as a conjecture. Establishing it for all n would show that the multiplicative Lie algebra structure captures all universal n-commutator identities in free groups without additional relations.