Conjecture on optimal cancellation for prime-power Gauss sums

Let $p$ be a prime and $k\ge 1$ an integer. Let $\chi\colon \mathbb{F}_q^\times\to\mathbb{C}^\times$ be a non-trivial character of order $o=\operatorname{ord}(\chi)$, where $o=p^k$, or $o=2p^k$ when $p$ is odd. The normalized prime-polynomial Gauss-sum sum is

$$\left|\sum_{\substack{\deg(\pi)=n\\ \pi\ \operatorname{prime}}} \frac{G_\chi(\pi)}{q^{n/2}}\right|=q^{\left(\frac12+\frac1{p^k}+o(1)\right)n}.$$

In the analogous number-field setting, the bias in order-$o$ Gauss sums is $(1+o(1))X^{\frac12+\frac1{p^k}}/\log X$. Optimal-cancellation conjecture. The stated asymptotics hold for the function-field and corresponding number-field Gauss-sum biases under these order conditions. The conjecture asserts that the exponent in the proven upper bound is optimal, while the paper notes that the separate factor $2^{2n-1}$ from the Betti-number estimate may be improvable because homology should concentrate away from the vanishing slope.