Neftin's simultaneous conjugation conjecture for permutations

Let $\alpha$ and $\beta$ be permutations in $S_n$. Write $[\alpha,\beta]=\alpha\beta\alpha^{-1}\beta^{-1}$ for their commutator, and let $\alpha^\gamma=\gamma^{-1}\alpha\gamma$ denote conjugation by $\gamma$. Neftin's conjecture. If $[\alpha,\beta]$ has at least $n-4$ fixed points, then there exists a permutation $\gamma\in S_n$ such that

$$\alpha^\gamma=\alpha^{-1}\qquad\text{and}\qquad\beta^\gamma=\beta^{-1}.$$

A positive solution would settle the hardest remaining cases in the primitive Hurwitz problem for $S_d\wr S_2$ and complete the Hurwitz part of the classification of monodromy groups for sufficiently large degrees; the parser reports that the conjecture has been resolved.