The periodic-point avoidance conjecture for consecutive-pattern-avoiding stack-sorting maps

Let $\sigma\in S_k$ with $k\geq 3$. For a permutation, $\operatorname{rev}(\sigma)$ denotes the reversal of $\sigma$, and underlined patterns indicate consecutive-pattern avoidance. The map $\operatorname{SC}_\sigma:S_n\to S_n$ is the consecutive-pattern-avoiding stack-sorting map.

Periodic-point avoidance conjecture. The periodic points of $\operatorname{SC}_\sigma$ are precisely the permutations in

$$\operatorname{Av}_n\left(\underline{\sigma},\underline{\operatorname{rev}(\sigma)}\right).$$

The claim is confirmed in the paper when $k=4$ and $n\leq 8$; the general assertion remains open.