Incidence bound for circles of different radii

Let $I_C(m,n)$ denote the maximum number of incidences between $m$ points and $n$ circles. The circle-incidence conjecture. For some positive constant $c$,

$$I_C(m,n)=O(m^{2/3}n^{2/3}\log^c(mn)+m+n).$$

This is a well-known incidence conjecture for circles of different radii; the paper cites Solymosi as an example of an external reference. Its status is not resolved in the supplied text.