Two-cycle count conjectures from the auxiliary heuristic

Let $p$ be prime, and let $T_{g\,A,h\,B}(p)$ count two-cycles with the restrictions indicated by $A$ and $B$. Let $\phi$ be Euler's totient function.

Two-cycle count conjectures.

$$T_{g\,\mathsf{PR},h\,\mathsf{ANY}}(p)\approx 2\phi(p-1),$$ $$T_{g\,\mathsf{ANY},h\,\mathsf{ANY}}(p)\approx 2(p-1).$$

The source derives these predictions using a heuristic relating the auxiliary equation to two-cycles; no resolution is given.