The cycle-membership conjecture for exact skew-symmetric plane permutations

Let $[n]^*$ and $[n]^{-}$ denote the signed sets used for the plane permutation, let $p_r$ be the reversal permutation, and let $D_{\mathfrak{p}}$ denote the diagonal permutation of the plane permutation $\mathfrak{p}=(\tilde{s},\pi)$. Assume that $\mathfrak{p}$ is exact and skew-symmetric on $[n]^*\cup[n]^{-}$, that $D_{\mathfrak{p}}=p_r^{-1}$, and that

$$\pi(s_{i-1})=s_{2n+1-i}$$

for some $1\leq i\leq n$.

Cycle-membership conjecture. The elements $n$ and $s_n$ belong to the same cycle of $\pi$.

This conjecture addresses the critical case in the analysis of 2-reversals for signed permutations, where the standard lemma does not apply. Establishing the asserted cycle membership would allow the use of the alternative transpose lemma to obtain a 2-reversal in this remaining case; the supplied text gives examples but no proof or resolution.