Uniform fourth-root bound conjecture for incomplete power sums

Let $p$ be prime and define

$$S(a,k,M)=\sum_{s=1}^{M}e\left(\frac{a s^k}{p}\right).$$

Here $a$ is understood modulo $p$, $k\in\mathbb{Z}_p^\times$, and $M$ is in the range used for incomplete sums in the paper. Uniform power-sum conjecture. For all $k\in\mathbb{Z}_p^\times$ with $(k,p-1)=1$ and $k\geq2$,

$$S(a,k,M)\ll p^{3/4},$$

uniformly in $a$, $k$, and $M$. Such a bound would imply quasirandomness of the associated power permutations and is stated as the paper's most intriguing conjecture, based on extensive computational evidence.