Ultra-flatness conjecture at roots of unity

Let $n>4$, and let

$$p(z)=a_0+a_1z+\dots+a_{n-1}z^{n-1}$$

be a polynomial with coefficients $a_i\in\{-1,1\}$. Ultra-flatness conjecture at roots of unity. There exists $\varepsilon_0>0$ such that

$$\max_{0\leq j\leq n-1}\left|\left|p\left(e^{\frac{2\pi i j}{n}}\right)\right|-\sqrt n\right|\geq\varepsilon_0 n^{1/4}.$$

This is a sampling analogue of the open problem of ultra-flat $\pm1$ polynomials: flatness is required only at the $n$th roots of unity. The source presents this as an open conjecture.