Sign changes of the partial sums of a random multiplicative function II

We study two models of random multiplicative functions: Rademacher random multiplicative functions supported on the squarefree integers $f$, and Rademacher random completely multiplicative functions $f^{\ast }$. We prove that the partial sums ${\sum }_{n\le x}{f}^{\ast }(n)$ and ${\sum }_{n\le x}\frac{f(n)}{\sqrt{n}}$ change sign infinitely often as $x\to \mathrm{\infty }$, almost surely. The case ${\sum }_{n\le x}\frac{f^{\ast }(n)}{\sqrt{n}}$ is left as an open question and we stress the possibility of only a finite number of sign changes, with positive probability.