H. Cohn's finite-field correlation conjecture

Let $p$ be an odd prime, and let $f:\mathbb{F}_p\to\mathbb{C}$ satisfy $f(0)=0$, $f(1)=1$, and $|f(a)|=1$ for every $a\in\mathbb{F}_p$. Suppose that, for every $h\in\mathbb{F}_p$,

$$\sum_{a\in\mathbb{F}_p}f(a)\overline{f(a+h)}=\begin{cases}-1&\text{if }h\ne0,\p-1&\text{otherwise.}\end{cases}$$

H. Cohn's conjecture. Then $f$ is a multiplicative character on $\mathbb{F}_p$. This problem is still open in the finite-field setting; partial results are known, but the conjecture remains unresolved.