Improvement on the Hanson–Petridis bound via degree-2 localization

Let $p$ be prime, let $G_p$ be the Paley graph, let $g$ be a multiplicative generator of $\mathbb F_p^\times$, and let $\vartheta$ denote the Lovász theta function. Improvement conjecture. There exists $\varepsilon>0$ such that, for all sufficiently large $p$,

$$\vartheta(G_{p,\{0,g\}})\leq\left(\frac1{\sqrt2}-\varepsilon\right)\sqrt p.$$

Here $G_{p,\{0,g\}}$ is the degree-2 localized Paley subgraph. Numerical experiments suggest an improvement over the Hanson–Petridis constant, but the assertion is not proved in the source.