Odd-moment equidistribution conjecture for quartic Gauss sums

For odd integers $k$, consider primitive quartic Dirichlet characters $\chi$ of odd conductor $q$, and let $\tau(\chi)$ denote their Gauss sums. Odd-moment equidistribution conjecture. For each odd $k$, there exists $\delta>0$ such that

$$\sum_{\substack{q\leq X\\(q,2)=1}}\ \sum_{\substack{\chi:\ \chi^4=1\operatorname{cond}(\chi)=q}}\left(\frac{\tau(\chi)^2}{q}\right)^k\ll_{k,\delta}X^{1-\delta}.$$

This estimate would provide the missing odd-moment input for Weyl's criterion and hence support equidistribution of the normalized squared quartic Gauss sums in the full quartic family. The even-moment analysis in the paper does not establish this odd-$k$ estimate.