On simultaneous conjugation of permutations

Let α and β be two permutations in $S_n$. We prove that if the commutator $[\alpha ,\beta ]$ has at least $n?4$ fixed points then there exists a permutation $\gamma \in S_n$ such that ${\alpha }^{\gamma }={\alpha }^{?1}$ and ${\beta }^{\gamma }={\beta }^{?1}$. This gives an affirmative answer to a conjecture proposed by Danny Neftin, which leads to the completion of the classification of monodromy groups for sufficiently large degree.