Cycle conjecture for exact skew-symmetric permutations

Let $\mathfrak{p}=(\tilde{s},\pi)$ be exact and skew-symmetric on $[n]^*\cup[n]^{-}$, with $D_{\mathfrak{p}}=p_r^{-1}$, and suppose that

$$\pi(s_{i-1})=s_{2n+1-i},\qquad 1\leq i\leq n.$$

Cycle conjecture. The elements $n$ and $s_n$ are in the same cycle of $\pi$.

This conjecture is introduced to provide a $2$-reversal in the critical case arising in the analysis of sorting signed permutations. The paper presents examples supporting it, but does not establish the claim; a broader version is stated later without the exactness assumption.