The pseudovariety decomposition conjecture for partial order-preserving permutations

Let $\mathcal{PODI}_n$ be the monoid of partial order-preserving or order-reversing permutations, let $\mathcal{POI}_n$ be the monoid of partial order-preserving permutations, and let $\mathcal{C}_2$ be the cyclic group of order two. Write $\pv{PODI}$ and $\pv{POI}$ for the pseudovarieties generated by the monoids $\mathcal{PODI}_n$ and $\mathcal{POI}_n$, respectively, and let $\pv{Ab_2}$ denote the pseudovariety generated by the groups $\mathcal{C}_2$. The notation $\!\rtimes\!$ denotes the bilateral semidirect product of pseudovarieties. Pseudovariety decomposition conjecture.

$$\pv{PODI}=\pv{POI}\!\rtimes\!\pv{Ab_2}.$$

The preceding results establish the corresponding inclusion in one direction and show that each finite monoid $\mathcal{POI}_n\!\rtimes\!\mathcal{C}_2$ belongs to $\pv{PODI}$; the conjecture asserts equality of the resulting pseudovarieties.