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

Let $p$ be a sufficiently large prime, let $m$ be a positive integer, and let $n$ be an integer satisfying $(n,p)=1$. Let $\chi_0$ be the principal character modulo $p$, let $L(s,\chi)$ be the associated Dirichlet $L$-function, and set $e(y)=e^{2\pi iy}$. Bag, Li and Zhang's conjecture.

$$\frac{1}{p^{m+1}}\sum_{\chi\neq\chi_0}\left|\sum_{a=1}^{p-1}\chi(a)e\left(\frac{na^2}{p}\right)\right|^{2m}|L(1,\chi)|=\binom{2m-1}{m}C+o(1),$$

where $C$ is the constant defined by the corresponding Euler product in the source. This conjecture concerns moments weighted by central values of Dirichlet $L$-functions. The supplied text does not state a proof of the general case, so it remains open here.