Counting sign changes of partial sums of random multiplicative functions

Let $f$ be a Rademacher random multiplicative function. Let $$M_f(u):=\sum _{n\le u}f(n)$$ be the partial sum of $f$. Let $V_f(x)$ denote the number of sign changes of $M_f(u)$ up to $x$. We show that for any constant $c>2$, $$V_f(x)=\mathrm{\Omega }((\log \log \log x{)}^{1\mathrm{/}c})$$ almost surely.