Paley graph conjecture

Let $p$ be a prime. For $0<\alpha\leq1$ and $\beta>0$, define the property $\mathcal{P}(\alpha,\beta)$ to mean that for every pair of subsets $S,T\subset\mathbb{F}_p$ with $|S|,|T|>p^\alpha$,

$$\left|\sum_{s\in S,\,t\in T}\chi(s-t)\right|\leq p^{-\beta}|S||T|.$$

Paley graph conjecture. For each $0<\alpha\leq1$, there exist $\beta=\beta(\alpha)>0$ and $p(\alpha)>0$ such that $\mathcal{P}(\alpha,\beta)$ holds for every prime $p>p(\alpha)$. This is a pseudorandomness conjecture for Paley graphs, asserting cancellation in quadratic-character sums between all sufficiently large vertex subsets. It is used in the paper to obtain the desired restricted isometry estimates for Paley matrices for both congruence classes of primes; the supplied text presents it as well known but gives no resolution, so its database status is open.