Asymptotic enumeration conjecture for plane curves

Let $a_p$ and $\alpha_p$ denote the numbers of open and closed plane curves, respectively, with $p$ self-intersections. Let $\gamma=-\frac{1+\sqrt{13}}6$. Asymptotic enumeration conjecture. There exist constants $\tau$ and $c$ such that

$$\alpha_p\mathop{\sim}_{p\to\infty}\frac14a_p\mathop{\sim}_{p\to\infty}c\,\tau^p\cdot p^{\gamma-2}.$$

This conjecture predicts both the exponential growth constant and the universal critical exponent for plane curves; the paper presents numerical evidence for it.