Uniform fixed-head divisor shift for friable counts

Fix a natural head cutoff $W$, and let $P_W$ be its rough-head modulus. There exist $K>0$ and $Y_0\in\mathbb N$ such that for all natural $X,y,d$ with $y\ge Y_0$, $d\mid P_W$, $d\le X$ and $\log X\le5\log y$,

Here $d$ ranges over the positive divisors of $P_W$ and $\Psi$ counts friable integers. The constants are uniform across the fixed finite divisor family.