$6$-th Norm of a Steinhaus Chaos

We prove that for the Steinhaus Random Variable $z(n)$ $$\mathbb{E}\left({\mid \sum _{n\in E_{N,m}}z(n)\mid }^6\right)\asymp \mathrm{\mid }{E}_{N,m}{\mathrm{\mid }}^3\text{ for }m\ll (\log \log N{)}^{\frac{1}{3}},$$ where $$E_{N,m}:=\{1\le n:\mathrm{\Omega }(n)=m\}$$ and $\mathrm{\Omega }(n)$ denotes the number of prime factors of $N$.