The asymptotic generating-triple count conjecture for prefix reversals

For $n$ and integers $m,k$ with $2\leqslant k<m<n$, let $f(n)$ be the number of pairs $(m,k)$ for which the prefix reversals $r_n,r_m,r_k$ generate the symmetric group $\operatorname{Sym}_n$. Generating-triple count conjecture. As $n$ tends to infinity,

$$f(n)=\begin{cases} \frac{1}{13}n^2+O(n)&n\equiv0\pmod{4},\\[6pt] \frac{1}{9}n^2+O(n)&n\equiv1\pmod{4},\\[6pt] \frac{1}{10}n^2+O(n)&n\equiv2\pmod{4},\\[6pt] \frac{1}{7}n^2+O(n)&n\equiv3\pmod{4}. \end{cases}$$

This is based on computational experiments for $6\leqslant n\leqslant70$; the source provides no proof or resolution.