Growth-rate conjecture for primitive Eisenstein circle packings

Let $E$) be a moiety of a bounded primitive Eisenstein circle packing. Let $N$ be a positive real number, and let $c_E$ denote a constant depending on $E$. Let $\delta\approx 1.4124$ be the Hausdorff dimension of the limit set of the packing. Growth-rate conjecture. The number of circles in $E$ with curvature at most $N$ is asymptotic to

$$c_E N^{\delta}.$$

This conjectural exponent is supported by computations and is analogous to the growth laws known for Apollonian circle packings and more general thin groups; the asymptotic remains unproved here.