Conjecture on the asymptotic growth of the excess normalized Gauss sums

For integers $n,q\geqslant 2$, define

$$S_n(a,q):=\sum_{x\bmod q}\mathbf{e}(ax^n/q), \qquad G_n(q):=\max_{\gcd(a,q)=1}|S_n(a,q)|,$$

and

$$A(n):=\sup_{q\geqslant 2}\frac{G_n(q)}{q^{1-1/n}}.$$

For $N\geqslant 2$, put

$$E(N):=\sum_{n=2}^N(A(n)-1).$$

Asymptotic-growth conjecture. There is a constant $c\in(1,3)$ such that

$$E(N)=(\log N)^{c+o(1)}\qquad(N\to\infty).$$

The paper proves only the bounds $(2+o(1))\log N\leqslant E(N)\ll(\log N)^3$, leaving the precise exponent open.