Zhang's moment conjecture for quadratic Gauss sums weighted by L-functions

Let $p$ be prime, let $n$ be an integer, and for a Dirichlet character $\chi\bmod p$ define the generalized quadratic Gauss sum

$$G(n,2,\chi;p)=\sum_{a=1}^{p}\chi(a)e\left(\frac{na^2}{p}\right),$$

where $e(y)=e^{2\pi i y}$. Let $\chi_0$ denote the principal character modulo $p$, let $m$ be a positive integer, and define

$$C=\prod_p\left[1+\frac{\binom{2}{1}^2}{4^2p^2}+\frac{\binom{4}{2}^2}{4^4p^4}+\cdots+\frac{\binom{2m}{m}^2}{4^{2m}p^{2m}}+\cdots\right],$$

where the product is over all primes. Zhang's conjecture. For all positive integers $m$,

$$\sum_{\chi\neq\chi_0}|G(n,2,\chi;p)|^{2m}|L(1,\chi)|\sim C\sum_{\chi\bmod p}|G(n,2,\chi;p)|^{2m},\qquad p\to+\infty.$$

This conjecture predicts an asymptotic relation between moments of quadratic Gauss sums weighted by the values of Dirichlet $L$-functions and the corresponding unweighted character sum. The source attributes it to Zhang and gives no evidence of a resolution.