Chowla's conjecture on quadratic character sums over large subsets

Let $0<\alpha\leq 1$ be a real number. Let $\mathbb{F}_p$ denote the finite field with $p$ elements, and let $\chi$ be the quadratic character modulo $p$. Chowla's conjecture. There exist $\beta=\beta(\alpha)>0$ and $p(\alpha)>0$ such that, for every prime $p>p(\alpha)$ and every subset $S\subset\mathbb{F}_p$ with $|S|>p^\alpha$,

$$\left|\sum_{s_1,s_2\in S}\chi(s_1-s_2)\right|\leq |S|^{2-\beta}.$$

This conjecture gives pseudorandomness estimates for quadratic residues on sufficiently large subsets of $\mathbb{F}_p$ and was used conditionally to establish the flat restricted isometry property of the Paley matrix in the case $p\equiv1\pmod 4$. It is trivial for $p\equiv3\pmod4$, because $\chi(-1)=-1$ makes the double sum vanish, but remains non-trivial in the other case.