Bag, Li and Zhang's generalized quadratic Gauss-sum moment conjecture

Let $p$ be a sufficiently large prime, let $m$ be a positive integer, let $n$ be an integer, and let $\chi$ range over Dirichlet characters modulo $p$. Set $e(y)=e^{2\pi iy}$. Bag, Li and Zhang's conjecture.

$$\frac{1}{p^{m+1}}\sum_{\chi\bmod p}\left|\sum_{a=1}^{p-1}\chi(a)e\left(\frac{na^2}{p}\right)\right|^{2m}=\binom{2m-1}{m}+o(1).$$

This is an asymptotic conjecture for the higher moments of generalized quadratic Gauss sums. The supplied text subsequently proves this statement, with an error term $O(p^{m+1/2})$, so its database status is solved.