Conjectures for fixed-point counts of the discrete logarithm map

Let $p$ be prime. For divisors and predicates appearing as subscripts, let $F_{g\,\mathsf{ANY},h\,\mathsf{ANY}}(p)$ denote the corresponding fixed-point count, and similarly for the restrictions $\mathsf{PR}$, $\mathsf{RP}$, and $\mathsf{RPPR}$; here $\phi$ is Euler's totient function and $\bullet$ denotes an unrestricted condition.

Fixed-point counting conjectures.

$$F_{g\,\mathsf{ANY},h\,\mathsf{ANY}}(p) \approx p-1,$$ $$F_{g\,\mathsf{PR},h\,\mathsf{ANY}}(p) \approx \phi(p-1),$$ $$F_{g\,\mathsf{RP},h\,\bullet}(p) \approx \frac{\phi(p-1)}{p-1}F_{g\,\mathsf{ANY},h\,\bullet}(p),$$ $$F_{g\,\mathsf{RPPR},h\,\bullet}(p) \approx \frac{\phi(p-1)}{p-1}F_{g\,\mathsf{PR},h\,\bullet}(p).$$

These conjectures refine heuristic independence assumptions about discrete logarithms and fixed points. Some special cases are stated in the surrounding text as theorems or are justified by the paper's heuristics, but the displayed asymptotic relations are presented as conjectures.