Necessity of separability for the strongly regular Cayley-graph characterization

Determine whether, for every integer n>4 and every Sidon set S⊆F_2^n of affine dimension n, S is a k-cover if and only if the Cayley graph Cay(γ_S) is strongly regular, without assuming that S is separable.

Background

The paper proves that, for n>4, a Sidon set S of affine dimension n is a k-cover if and only if Cay(γ_S) is strongly regular and S is separable. The authors explicitly leave unresolved whether the separability hypothesis is necessary, which would yield a characterization solely in terms of strong regularity of the Cayley graph.

The paper also gives an equivalent Fourier-analytic formulation involving whether the image of (\widehat{1_S})^2 has size three, but the strongly regular graph formulation is the clearest direct statement of the unresolved problem.