Short-interval conjecture for integers without small prime factors

Let $X$, $y$ and $H$ be parameters satisfying the conditions denoted by

$$and$$

, and let $\alpha_y(n)$ be the indicator of integers without prime factors at most $y$, with mean density $P_y$. Let $\Psi(H,y)$ denote the corresponding friable-number quantity. Short-interval conjecture. For all $X$, $y$ and $H$ satisfying

$$and$$

for some $\varepsilon,\delta>0$ we have

$$\sum_{X<n \le X+H} \alpha_y(n) = H P_y + O_{\varepsilon,\delta}\left( (Hy)^{\varepsilon}\Psi(H,y)^{1/2}\right).$$

This conjecture gives an essentially square-root error term for the count of integers without small prime factors in short intervals, extending the expected behavior suggested by analogous conjectures for primes. Its status is not resolved in the supplied source.