Average auxiliary-count conjecture

Let $p$ range over primes, let $C_{a\,\mathsf{ANY},h\,\mathsf{ANY}}(p)$ be the auxiliary solution count used in the paper, let $\operatorname{Li}(x)$ be the logarithmic integral, and let $C>0$ be arbitrary.

Average auxiliary-count conjecture.

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

The numerical constant is motivated by the observed average of an auxiliary arithmetic function. The supplied text gives no proof or resolution of this conjecture.