The asymptotic conjecture for connected functional graphs of quadratic polynomials

Let $p$ be a prime, and let $I_p$ denote the number of parameters $a$ for which the functional graph associated with the quadratic polynomial $f_a$ is connected. The computations use the connectedness-testing algorithm described above and the corresponding values of $I_p$. Asymptotic conjecture.

$$I_p \sim \sqrt{2p}\quad\text{as }p\to\infty.$$

This conjecture is based on computational data for selected primes and predicts the asymptotic growth of the number of connected functional graphs in this family. The supplied text gives no proof or resolution, so its status remains open.