Average-order conjecture for arbitrary two-cycle counts

Let $p$ range over primes, let $C_{h\,\mathop\mathsf{ANY},a\,\mathop\mathsf{ANY}}(p)$ be the number of solutions of the two-cycle equation with $h$ and $a$ unrestricted, and let $\operatorname{Li}(x)$ denote the logarithmic integral. For any fixed $C>0$, write $O_C(\cdot)$ for a bound whose implied constant may depend on $C$.

Average two-cycle conjecture. For every $C>0$,

$$\sum_{p\leq x}\frac{C_{h\,\mathop\mathsf{ANY},a\,\mathop\mathsf{ANY}}(p)}{p-1}=2.644\cdots\operatorname{Li}(x)+O_C\left(\frac{x}{\log^C x}\right).$$

This conjecture is motivated by a numerical average for an auxiliary arithmetic function and predicts an asymptotic formula for the prime average of the unrestricted two-cycle count. The source explicitly describes the assertion as conjectural and gives no proof or resolution.