  Let $f$ be a Rademacher or a Steinhaus random multiplicative function. Let
$\varepsilon>0$ small. We prove that, as $x\rightarrow +\infty$, we almost
surely have $$\bigg|\sum_{\substack{n\leq x\\
P(n)>\sqrt{x}}}f(n)\bigg|\leq\sqrt{x}(\log\log x)^{1/4+\varepsilon},$$ where
$P(n)$ stands for the largest prime factor of $n$. This gives an indication of
the almost sure size of the largest fluctuations of $f$.
