Relative-primitivity scaling conjectures for two-cycle counts

Let $p$ be prime, and let $T_{g\,A,h\,B}(p)$ denote the relevant two-cycle count, with $\bullet$ representing an unrestricted condition. Let $\phi$ be Euler's totient function.

Relative-primitivity scaling conjectures.

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

These relations are justified in the source by a heuristic concerning the auxiliary equation, but are not proved there.