One way to prove Erdős-Kac is by proving that for each non-negative integer $k$, the $k$-th moment $$M_k(x):= \frac{1}{x}\sum_{1 \leq n \leq x} \left(\frac{\omega(n)-\log \log x}{\sqrt{\log \log x}}\right)^k $$ converges as $x\to+\infty$ to the $k$-th moment of the standard normal distribution.
I was wondering whether the following weak converse is true, i.e. given the Erdős-Kac theorem can we prove that $M_k(x)=O_k(1)$ without number theory arguments?
