The linear-programming improvement conjecture for Paley graph clique bounds

Let $G_p$ be the Paley graph for a prime $p\equiv 1~(\operatorname{mod}4)$, let $\operatorname{LS}(p)$ denote the linear programming bound obtained in the paper, and let $\operatorname{HP}(p)$ denote the Hanson--Petridis bound. Linear-programming improvement conjecture. For infinitely many primes $p\equiv 1~(\operatorname{mod}4)$, one has

$$\operatorname{LS}(p)<\lfloor \operatorname{HP}(p)\rfloor.$$

The numerical experiments suggest that this linear programming bound sometimes improves on the best known closed-form Hanson--Petridis bound; proving infinitely many such improvements remains open.