Two-cycle count conjectures for the discrete logarithm map

Let $p$ be prime. Write $T_{g\,A,h\,B}(p)$ for the number of two-cycles with the indicated restrictions on $g$ and $h$, and $C_{h\,A,a\,B}(p)$ for the corresponding count in the auxiliary formulation; the symbols $\mathsf{ANY}$, $\mathsf{PR}$, $\mathsf{RP}$, $\mathsf{RPPR}$, $\mathsf{ORD}\,h$, and $\bullet$ have the meanings fixed earlier in the paper. Let $\phi$ be Euler's totient function.

Two-cycle counting conjectures.

$$T_{g\,\mathsf{ANY},h\,\mathsf{RP}}(p)=C_{h\,\mathsf{RP},a\,\mathsf{ANY}}(p)\approx 2\phi(p-1),$$ $$T_{h\,\mathsf{RP},g\,\mathsf{ORD}\,h}(p)=C_{h\,\mathsf{RP},a\,\mathsf{RP}}(p)\approx \phi(p-1)+\frac{\phi(p-1)^2}{p-1},$$ $$T_{g\,\mathsf{PR},h\,\mathsf{RP}}(p)=C_{h\,\mathsf{RP},a\,\mathsf{PR}}(p)\approx \frac{2\phi(p-1)^2}{p-1},$$ $$T_{g\,\mathsf{PR},h\,\mathsf{RPPR}}(p)=T_{g\,\mathsf{ANY},h\,\mathsf{RPPR}}(p)=C_{h\,\mathsf{RPPR},a\,\bullet}(p)=C_{h\,\bullet,a\,\mathsf{RPPR}}(p)\approx \frac{\phi(p-1)^2}{p-1}+\frac{\phi(p-1)^3}{(p-1)^2},$$ $$T_{h\,\mathsf{ANY},g\,\mathsf{ORD}\,h}(p)=C_{h\,\mathsf{ANY},a\,\mathsf{RP}}(p)\approx 2\phi(p-1),$$ $$T_{g\,\mathsf{PR},h\,\mathsf{PR}}(p)=T_{g\,\mathsf{ANY},h\,\mathsf{PR}}(p)=C_{h\,\mathsf{PR},a\,\mathsf{RP}}(p)\approx \frac{2\phi(p-1)^2}{p-1}.$$

The equalities in the first, fifth, third, and sixth relations are noted in the source to be exact by symmetry. The asymptotic predictions arise from a random-map heuristic and are not proved in the supplied text.