Polylogarithmic independence number conjecture for Paley graphs

For a prime $p$, let $G_p$ be the Paley graph, let $\overline G_p$ be its complement, and write $\alpha(G_p)=\omega(\overline G_p)$ for its independence number. Here $\mathsf{polylog}(p)$ denotes a fixed polylogarithmic function of $p$. Polylogarithmic independence-number conjecture.

$$\alpha(G_p)=O(\mathsf{polylog}(p)).$$

Determining the independence number is a central problem connected with bounds for quadratic non-residues and Ramsey theory. The statement is described as widely believed, but remains open in the source.