Replace the rational separation certificate with an analytic inequality

Construct a short analytic inequality, analogous to the pentagon Hilbert–Schmidt inequality, that proves a strict separation between \(\vartheta_1(G)\) and \(\log_2\Sigma(G)\) for a suitable graph.

Background

The strict separation for the 32-vertex magic-square graph is established using exact rational certificates from a moment relaxation.

The authors ask whether this computational certificate can be replaced by a concise analytic argument of the type used to prove the pentagon result.