Soundararajan's conjecture on smooth numbers in arithmetic progressions

Let $P(n)$ denote the largest prime factor of $n$, with $P(1)=1$, and call $n$ $y$-smooth when $P(n)\le y$. Write $\mathcal{S}(x,y)$ for the $y$-smooth numbers at most $x$, and define

$$\Psi(x,y;q,a)=\sum_{\substack{n\in\mathcal{S}(x,y)\\ n\equiv a\bmod q}}1,$$

and

$$\Psi_q(x,y)=\sum_{\substack{n\in\mathcal{S}(x,y)\\ (n,q)=1}}1.$$

For $(a,q)=1$, the expected equidistribution relation is

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

Soundararajan's conjecture. For any fixed value of $A>0$, if $q$ is sufficiently large, depending only on $A$, with $q\le y^A$ and $(a,q)=1$, then

$$\Psi(x,y;q,a)\sim\frac{\Psi_q(x,y)}{\varphi(q)}$$

as $\log x/\log q\to\infty$. The conjecture concerns the distribution of smooth numbers among reduced residue classes. Granville established it for $A<1$; the general assertion was proved in the paper, and its proof would imply Vinogradov's conjecture that the least quadratic nonresidue modulo $p$ has size $p^{o(1)}$.