Moment conjecture for Steinhaus random multiplicative functions

Let $f$ be a Steinhaus random multiplicative function: the values $f(p)$ are independent and uniformly distributed on the complex unit circle, and $f(n)=\prod_{p^{\alpha}\parallel n}f(p)^\alpha$. For $q\geq0$, define the random sum $S_N=\sum_{n\leq N}f(n)$ and let $C(q)$ denote the conjectural leading constant.

Random-multiplicative-function moment conjecture. As $N\rightarrow\infty$,

$$\mathbb{E}|S_N|^{2q}\sim \begin{cases} C(q)N^q,&0\leq q\leq1,\\ C(q)N^q(\log N)^{(q-1)^2},&1\leq q. \end{cases}$$

This is presented as a counter-conjecture to Helson's first-moment claim: at $q=1/2$, it predicts order $\sqrt N$ rather than $o(\sqrt N)$. It is motivated by the paper's moment asymptotics, while the full piecewise prediction remains conjectural.