The cost classification conjecture for skew-layered permutations

Let $\pi$ be a skew-layered permutation of size $n$, and let $\mathsf{cost}(\pi)$ denote its pop-stack sorting cost. Exclude $\mathsf{id}$ and $-\mathsf{id}$. Skew-layered cost classification conjecture. If $n$ is even, then

$$\mathsf{cost}(\pi)=n-1.$$

If $n$ is odd and the $(n+1)/2$-th position of $\pi$ is the central point of a run or of a fall of size at least $3$, then $\mathsf{cost}(\pi)=n-2$; otherwise, $\mathsf{cost}(\pi)=n-1$. This conjecture refines the preceding necessary condition for permutations of maximum cost and is based on the authors' computer experiments; a general characterization of all maximum-cost permutations remains open.