Periodic-point conjecture for three generalized stack-sorting maps

Let $S_n$ be the set of permutations of length $n$. For a pair of patterns $(\sigma,\tau)$, let $s_{\sigma,\tau}:S_n\to S_n$ denote the corresponding generalized stack-sorting map, and call a permutation periodic if some positive iterate of the map sends it to itself. Periodic-point conjecture. For

$$(\sigma,\tau)=(123,213),\ (132,312),\ (231,321),$$

the map $s_{\sigma,\tau}$ is a bijection from $S_n$ to itself, and all permutations are periodic. This conjecture concerns the dynamics of generalized stack-sorting maps; its status is not specified in the source.