Derivability of higher universal commutator identities

Establish, for every n≥3 and every free group G, the equality ⟨φ_n^{-1}(K_3(G))⟩=C_n(G), where φ_n maps the free group on simple n-fold bracket arrangements to S_n(G) and C_n(G) is the group of all n-commutator relations; equivalently, prove that all basic universal n-commutator identities are derivable from the five fundamental commutator identities.

Background

The groups C_n(G) and B_n(G) represent, respectively, all n-commutator relations and universal n-commutator relations. For n=3, the set K_3 consists of relations generated from the defining identities of multiplicative Lie algebras together with a group-theoretic Jacobi-type identity.

For n≥3, the paper constructs a map φ_n into the subgroup S_n(G) of the non-abelian exterior square G∧G and observes that the subgroup generated by the preimage of K_3(G) is contained in C_n(G). The unresolved issue is whether this inclusion is an equality for free groups, which would imply that no additional basic universal identities are needed beyond the five listed fundamental identities.

References

Is it true that ${ [a_1,b_1] \ldots [a_n, b_n], g_2, \ldots, g_{n-1} } \in \ker(\pi_n)$?

— The free multiplicative Lie algebra $L(P)$ for a finitely generated parafree group $P$  (2609.19595 - Kochloukova, 17 Sep 2026) in Section 'More generalisations', unfinished proof for orientable surface groups

Clearly, < \phi_{n}{-1}(K_{3}(G)) >\subseteq C_{n}(G), and it is conjectured that the equality holds for all $n\geq 3$, whenever $G$ is free group ().

— Commutator identities, Lie product identities, and Multiplicative Lie algebras  (2608.30998 - Kakkar et al., 31 Aug 2026) in Section 2, subsection beginning “Next, we recall some basics in multiplicative Lie algebras” (discussion following the definition of S_n(G))