Uniform better-than-square-root cancellation conjecture for smooth numbers

Let $f$ be a Steinhaus random multiplicative function, let $P(n)$ denote the largest prime factor of $n$, and define

$$\Psi(x,y)=\#\{n\leqslant x:P(n)\leqslant y\}.$$

Uniform smooth-number cancellation conjecture. The paper conjectures that its main estimate should hold uniformly for every $2\leqslant y\leqslant x$:

$$\mathbb E\left|\sum_{\substack{1\leqslant n\leqslant x\P(n)\leqslant y}}f(n)\right|=o\left(\Psi(x,y)^{1/2}\right).$$

The proved theorem establishes this only for $(\log x)^{30}\leqslant y\leqslant x$; extending it to the full range $2\leqslant y\leqslant x$ remains open.