Helson's conjecture for smooth numbers

Let $Ψ(x,y)$ 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 enjoy better than squareroot cancellation, in the sense that E |∑_1≤ n ≤ x\\ P(n) ≤ y f(n) | = o( Ψ(x,y)1/2 ), uniformly for (log x)³⁰ ≤ y ≤ x. Our bounds are quantitative and give a large saving when y isn't too close to x.