Average fixed-point count conjectures

Let $F_{g\,A,h\,B}(p)$ denote the fixed-point count for prime $p$, let $\operatorname{Li}(x)$ be the logarithmic integral, and let $A_1$ be the constant used in the paper.

Average fixed-point conjectures.

$$\sum_{p\le x}\frac{F_{g\,\mathsf{ANY},h\,\mathsf{ANY}}(p)}{p-1}\approx \operatorname{Li}(x),$$ $$\sum_{p\le x}\frac{F_{g\,\mathsf{PR},h\,\mathsf{ANY}}(p)}{p-1}\approx A_1\operatorname{Li}(x).$$

These are average versions of the fixed-point conjectures. The source presents them as conjectural numerical asymptotics and supplies no proof in the given passage.