The periodic-point conjecture for the stack-sorting maps $s_{\{132,213\}}$ and $s_{\{231,213\}}$

Let $S_n$ be the set of permutations of length $n$, and let $s_{\{132,213\}}$ and $s_{\{231,213\}}$ be the corresponding stack-sorting maps. A permutation is periodic if some positive iterate of the relevant map sends it back to itself. Periodic-point conjecture. The only periodic points of $s_{\{132,213\}}$ and $s_{\{231,213\}}$ are the identity permutation and its reverse. This conjecture concerns the eventual periodic behavior of these restricted stack-sorting maps; the preceding results determine periodic points for related maps, while the asserted classification for these two maps remains unresolved in the source.