Asymptotic equivalence of A2k and A2k+1

Ascertain whether, for k ≥ 2, the varieties A2k and A2k+1 of Lie-nilpotent associative algebras (of indexes 2k and 2k+1, respectively) are asymptotically equivalent, i.e., whether they satisfy the same proper multilinear identities in all sufficiently large degrees.

Background

The authors define asymptotic equivalence of varieties as agreement on all proper polynomial identities in sufficiently large degrees. They prove that A2 and A3 are asymptotically equivalent (Proposition 6.1), while A2k−1 and A2k are not asymptotically equivalent for any k ≥ 1 (Proposition 6.2).

Motivated by these contrasting results, they pose whether the specific pair A2k and A2k+1 share asymptotic equivalence for k ≥ 2, extending the A2/A3 phenomenon to higher indices.