Finiteness of higher-nilpotency Mal'cev algebras on squarefree sets

Determine whether, for every squarefree positive integer n and every integer k at least 3, the number of k-nilpotent Mal'cev algebras on the set {1,2,\ldots,n}, up to term equivalence, is finite.

Background

The paper proves that the number of 2-nilpotent Mal'cev algebras on an n-element set, up to term equivalence, is finite exactly when n is squarefree. For non-squarefree n, countably infinitely many such algebras exist.

The stated question asks whether the finiteness phenomenon persists for nilpotency levels k at least 3 when the underlying set has squarefree cardinality. The paper notes that answering it by analogous methods would require a better understanding of clonoids from nilpotent non-abelian Mal'cev algebras into abelian Mal'cev algebras.