Etingof–Kim–Ma conjecture on products of commutator ideals

Let $F$ be a field of characteristic $0$, let $F\langle X\rangle$ be the free unital associative algebra on a countable set, and for $n\geq 2$ let $T^{(n)}$ be the two-sided ideal generated by all left-normed $n$-fold commutators.

Etingof–Kim–Ma conjecture.

$$T^{(m)}T^{(n)}\subset T^{(m+n-1)}$$

if and only if $m$ or $n$ is odd.

The conjecture was confirmed in the source paper: Bapat and Jordan had already proved the “if” direction, and the paper proves the “only if” direction by showing that the inclusion fails when both $m$ and $n$ are even. Thus the conjecture is solved.