Sign changes on $\sum \frac{f(n)}{\sqrt{n}}$ for random completely multiplicative $f$

If $f: \mathbb{N} \rightarrow \{\pm1\}$ is a sample of the random completely multiplicative function, we show that almost surely $\sum_{n \le x} \frac{f(n)}{\sqrt{n}}$ changes signs infinitely many times, answering a question of Aymone.