The cycle-membership conjecture for 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 skew-symmetric on $[n]^*\cup[n]^{-}$ and that

$$D_{\mathfrak{p}}=p_r^{-1}.$$

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

This is presented as a generalization of the preceding exact-case conjecture and is intended to support the comparison between the paper's lower bound for signed reversal distance and the break-point-graph bound. The supplied text gives no proof or resolution.