The intersection bound for quadratic graph parameter sets

Let $p\geq3$ be prime. Let ${\mathscr A}_p$ and ${\mathscr B}_p^*$ be the parameter sets defined in the paper, with ${\mathscr A}_p$ denoting the relevant set of parameters and ${\mathscr B}_p^*$ the subset associated with connected functional graphs. The intersection-bound conjecture.

$$\#\left({\mathscr A}_p\cap{\mathscr B}_p^*\right)\leq1.$$

Computations found only one parameter in the intersection for each observed prime, but the bound is not proved in general.