Helson's conjecture for smooth numbers

Let $\mathrm{\Psi }(x,y)$ denote the count of $y$-smooth numbers below $x$ and $P(n)$ denote the largest prime factor of $n$. We prove that for $f$ a Steinhaus random multiplicative function, the partial sums over $y$-smooth numbers always enjoy better than squareroot cancellation, in the sense that $$\mathbb{E}\mid \sum _{\begin{array}{c}1\le n\le x\\ P(n)\le y\end{array}}f(n)\mid =o(\mathrm{\Psi }(x,y{)}^{1\mathrm{/}2}),$$ uniformly on the entire range $2\le y\le x$. The bounds are quantitative and give a large saving when $y$ isn’t too close to $x$.