Refined stabilization criterion for the upper-central series

Develop a refined characterization, for perfect semiassociative Mal'cev algebras that do not satisfy an unconditional Grün's lemma, of when the upper-central series stabilizes, particularly of when the second-center coincides with the center.

Background

Theorem 3 establishes a general conditional criterion for equality of the center and second-center using generating sets of binary absorbing polynomials. The paper notes that special algebraic identities can yield sharper criteria, as illustrated for perfect skew braces. The open problem seeks analogous refined criteria for all perfect semiassociative Mal'cev algebras in which unconditional stabilization fails.