The stabilization-time conjecture for the map \operatorname{SC}_{231}

Let $n\geq 3$, let $S_n$ be the set of permutations of $[n]$, and let $\operatorname{Av}_n(132,231)$ denote the permutations avoiding $132$ and $231$.

Stabilization-time conjecture. For every $\pi\in S_n$,

$$\operatorname{SC}_{231}^{2n-4}(\pi)\in\operatorname{Av}_n(132,231).$$

Furthermore, there exists $\tau\in S_n$ such that

$$\operatorname{SC}_{231}^{2n-5}(\tau)\notin\operatorname{Av}_n(132,231).$$

The conjecture has been verified for $n\leq 9$ and predicts the sharp number of iterations needed to reach the periodic-point set for this map.