Amenability converse for central Fourier algebras

Determine whether amenability of the central Fourier algebra $ZA(G)$ for a compact group $G$ implies that $G$ is virtually abelian, thereby establishing the converse to the known sufficient condition.

Background

The paper summarizes known results showing that $ZA(G)$ is amenable whenever a compact group $G$ is virtually abelian, meaning that $G$ has an abelian subgroup of finite index. Partial converses are known for connected compact groups and for infinite products of finite non-abelian groups, but the general compact-group case is left unresolved. The same conjecture is attributed to Alaghmandan and Spronk in the discussion of the central Fourier algebra.