Existence of connected quadratic functional graphs modulo every prime

Let $p$ be prime, and let $I_p$ denote the number of parameters $a\in\mathbb F_p$ for which the functional graph ${\mathcal G}_a$ of $f_a(X)=X^2+a$ is connected. The existence conjecture. For any prime $p$,

$$I_p\geq 1.$$

This weaker conjecture predicts that every prime admits at least one connected functional graph generated by a quadratic polynomial; the authors report verification for all primes $p\leq100000$.