Large fluctuations of extended Rademacher random multiplicative functions

Let $f$ be an extended Rademacher random multiplicative function (RMF). We show that, for every fixed deterministic function $V(x)$ tending to infinity, almost surely there are arbitrarily large $x$ for which $$\sum _{n\le x}f(n)\ge \frac{\sqrt{x}(\log \log x{)}^{1\mathrm{/}4}}{V(x)}\mathrm{.}$$ The corresponding negative fluctuation holds as well. In particular, this gives an affirmative answer to Erdős Problem~#1144. As a byproduct, our result has a direct corollary giving new almost sure lower bounds $\log \log x$ on the number of sign changes of partial sums up to $x$ for all sufficiently large $x$.