Moments of random multiplicative functions with polynomial coefficients

Let $f$ be a Steinhaus random multiplicative function and let $g$ be a polynomial of degree $d$. Write $S_N=N^{-1/2}\sum_{n\le N}f(n)\,e(g(n))$. We prove a quantitative dichotomy for the moments of $S_N$: for each integer $s\ge 2$, either $\mathbb{E}|S_N|^{2s}$ is close to the Gaussian moment $s!$,