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 ₍ ₗ\\ ₏ (₍) ₘ f (n) | = o (Ψ (x, y) ^1/2), uniformly for (x) ^30 y x. Our bounds are quantitative and give a large saving when y isn't too close to x.