Alternating-group monodromy conjecture for minimal [1,1]-origamis

A minimal $[1,1]$-origami in $\mathcal{H}(2g-2)$ has $2g-1$ squares, and its monodromy group is a subgroup of the alternating group on these squares. The $[1,1]$-origamis considered in Theorem~ are the minimal $[1,1]$-origamis constructed by Aougab-Menasco-Nieland and their generalisations.

Alternating-group monodromy conjecture. All of the $[1,1]$-origamis in Theorem~ have monodromy group isomorphic to $\operatorname{Alt}_{2g-1}$.

The theorem cited in the statement leaves open whether the monodromy group is alternating or projective special linear; this conjecture, based on computer experiments, predicts that the alternating case always occurs. The number of squares, $2g-1$, lies below the threshold in a result cited by the authors that would force the monodromy group to contain the alternating group.