The \operatorname{SC}_{132} image conjecture for maximally nonperiodic permutations

Let $s_{132}$ be the classical stack-sorting map associated with the pattern $132$, let $\operatorname{Av}(132,231)$ denote the permutations avoiding $132$ and $231$, and define ${\bf V}_n$ by listing the odd elements of $[n]$ in decreasing order followed by the even elements in increasing order when $n$ is odd, with the analogous even-then-odd construction when $n$ is even.

\operatorname{SC}_{132} image conjecture. If $\tau\in S_n$ satisfies

$$s_{132}^{n-2}(\tau)\notin\operatorname{Av}(132,231),$$

then

$$s_{132}^{n-1}(\tau)={\bf V}_n.$$

The claim concerns permutations whose second-to-last iterate has not yet entered the known periodic-point set; the paper presents it as an additional open dynamical question.