Chung's Paley discrepancy conjecture

For primes $p\equiv1\bmod4$, let $\chi$ be the Legendre symbol modulo $p$. Define $\operatorname{PaleyDiscrepancy}[\alpha,\beta]$ to mean that, for every sufficiently large $p$,

$$\left|\sum_{a,b\in S}\chi(a-b)\right|<|S|^{2-\beta}$$

for every $S\subseteq\mathbb{F}_p$ with $|S|>p^\alpha$. Chung's conjecture. For each $\alpha>0$, there exists $\beta=\beta(\alpha)>0$ such that $\operatorname{PaleyDiscrepancy}[\alpha,\beta]$. This folklore conjecture appeared as Conjecture 2.2 in Chung (1994), with a proof for $\alpha>1/2$ given there, and has also been used in constructing a pseudorandom number generator. The source does not state a complete resolution for arbitrary $\alpha>0$.