Stability of identities following from associativity under ultrafilter algebras

Let an identity be a formula in the language of a groupoid, and consider the algebra of ultrafilters associated with a compact left topological groupoid. Ultrafilter-stability conjecture. Identities that follow from associativity are stable under passing to the algebra of ultrafilters. This asks which formulas, or at least which identities, are preserved by the ultrafilter construction; the source provides no resolution of the conjecture.