Uniform multiplication-table congruence bound

Let $\delta\in(0,1)$ be fixed, let $q$ and $c$ be integers with $(c,q)=1$, and let $N$ be positive. Uniform multiplication-table congruence conjecture. One has

$$\#\{(u,v)\in\mathbb{N}^2:u,v\le N,\ uv\equiv c\pmod q\}\ll_{\delta}\phi(q)q^{-2}N^2$$

uniformly for $q\le N^{2(1-\delta)}$. Removing the logarithmic factor from the known Linnik–Vinogradov estimate would improve the analysis of the pair correlation function, but the conjectured bound is not proved in the source.