Average-size conjecture for sporadic square divisibility classes

For each prime $p$, let ${\mathcal C}_p$ be the set defined in the paper that records the relevant sporadic roots or residue classes. Average-size conjecture for ${\mathcal C}_p$. The sets ${\mathcal C}_p$ have one element on average over the primes, in the sense that

$$\sum_{p<x}\#{\mathcal C}_p\sim\pi(x).$$

This heuristic is used to explain the observed density calculation for squarefree values, but the asserted asymptotic is not proved in the paper.