The smooth numbers conjecture in arithmetic progressions

Let $A$ be a given positive real number. For positive integers $x,y,a,q$, let $\Psi(x,y,a,q)$ count the $y$-smooth positive integers less than $x$ that are congruent to $a$ modulo $q$, and let $\Psi_q(x,y)$ count those coprime to $q$. Here $\phi(q)$ is Euler's totient function.

Smooth numbers conjecture. If $y$ and $q$ are large with $q\leq y^A$, then, as $\frac{\log x}{\log y}\rightarrow\infty$,

$$\Psi(x,y,a,q)\sim \frac{1}{\phi(q)}\Psi_q(x,y).$$

This conjecture predicts that sufficiently large moduli up to a fixed power of the smoothness bound do not bias the distribution of smooth numbers among reduced residue classes. The source presents it as a central goal and attributes it to Soundararajan; no resolution is given here.