Let $c_1,c_2\in S_n$ be two disjoint cycles of length $|c_1|$ and $|c_2|$ respectively. Let $I(c_i)$ be the coordinates on which permutation $c_i$ acts at $i\in\{1,2\}$. Note by choice we have $I(c_1)\cap I(c_2)=\emptyset$.
Pick $\sigma\in S_n$ such that $I(\sigma)\supseteq I(c_1)\cup I(c_2)$.
What is the probability that $c_1\circ c_2\circ\sigma=d_1\circ d_2\circ\pi$ where $d_1$ and $d_2$ are disjoint cycles with $|d_i|=|c_i|$ and $I(c_i)=I(d_i)$ at $i\in\{1,2\}$ holds and $\pi\in S_n$ is a cycle disjoint from $d_1$ and $d_2$ with $I(\pi)=I(\sigma)\backslash(I(c_1)\cup I(c_2))$?
Is this $\frac{(|c_1|-1)!(|c_2|-1)!}{(|c_1|+|c_2|)!}$ or something much smaller?
What is the probability that $c_1\circ c_2\circ\sigma=\rho\circ\pi$ where $\rho\in S_n$ is a permutation with $I(\rho)=I(c_1)\cup I(c_2)$ and $\pi\in S_n$ is a cycle disjoint from $\rho$ with $I(\pi)=I(\sigma)\backslash I(\rho)$?
Note: If $I(\sigma)=I(c_1)\cup I(c_2)$ then $\pi$ is empty.