Four-root-of-unity anti-concentration conjecture

Let $\xi_1,\xi_2,\xi_3,\xi_4$ be independent random variables, each uniformly distributed on the set of $n$th roots of unity.

Four-root-of-unity anti-concentration conjecture. As $n\to\infty$,

$$\mathbb{P}\left(\left|\operatorname{Re}(\xi_1+\xi_2+\xi_3+\xi_4)\right|<\frac{1}{n^2}\right)=\frac{1}{n^{2+o(1)}}.$$

The conjecture is presented as an obstacle to proving regularity of the unperturbed eigenvalues without uniform boundary conditions. The source gives no resolution, so its status remains open.