SL(2,Z)-orbit conjecture for the constructed origamis

Origamis are translation surfaces on square-tiled coverings, and the group $\operatorname{SL}(2,\mathbb{Z})$ acts on them while preserving the connected components of strata. The conjecture concerns the Aougab-Menasco-Nieland origamis and the odd- and even-genus generalisations specified below.

SL(2,Z)-orbit conjecture. For odd genus $g\geq 5$, the origamis constructed by Aougab-Menasco-Nieland lie in exactly two $\operatorname{SL}(2,\mathbb{Z})$-orbits inside the odd component; in genus three there is only one orbit because the construction gives a single origami. The odd genus generalisations of Subsection~ lie in a single $\operatorname{SL}(2,\mathbb{Z})$-orbit inside the even component. The even genus Aougab-Menasco-Nieland origamis and the even genus generalisations of Section~ lie in a single $\operatorname{SL}(2,\mathbb{Z})$-orbit in each of the odd and even components.

The theorem on orientation double covers implies that the odd genus Aougab-Menasco-Nieland origamis lie in at least two $\operatorname{SL}(2,\mathbb{Z})$-orbits for $g\geq 5$, while the conjecture gives the precise orbit counts and predicts transitivity for the other listed families within the indicated components. The final paragraph means that constructions lying in the same connected component are in fact contained in the same orbit.