Degree-4 Sum-of-Squares improvement for Paley clique number

Show that there exists ε > 0 such that the degree-4 Sum-of-Squares relaxation for the clique number of the Paley graph yields an O(p^{1/2 − ε}) upper bound.

Background

The Lovasz theta (degree-2 SoS) yields at best √p-type bounds for Paley graphs. Numerical evidence suggests that degree-4 SoS could achieve better-than-√p scaling, potentially polynomially better, though lower bounds limit improvements to Ω(p^{1/3}).

Proving an O(p^{1/2−ε}) bound would demonstrate a concrete advantage of low-degree SoS over classical SDP relaxations for Paley graphs.