Covering an integer grid by circles of distinct radii

Let $n$ be a positive integer, and consider the $n\times n$ grid. Suppose it is covered by $m$ circles, with no two circles having equal radius. Distinct-radii circle-covering conjecture. For some positive constant $c$,

$$m=\Omega\left(\frac{n^2}{\log^c(n)}\right).$$

The paper states this as a covering consequence of the preceding circle-incidence conjecture. The supplied text does not establish the bound, so its resolution remains open.