Super-exponential growth of minimal-origami orbits

Let $N_g$ denote the number of distinct $SL(2,\mathbb{Z})$-orbits of origamis in $\mathcal{H}(2g-2)$ with $2g-1$ squares, such that each of their horizontal and vertical cylinder decompositions consists of one cylinder. Super-exponential orbit-growth conjecture. The quantity $N_g$ grows super-exponentially in $g$. These origamis, called minimal origamis, have the simplest possible horizontal and vertical cylinder decompositions and the smallest possible number of squares. The conjecture predicts that the number of distinct $SL(2,\mathbb{Z})$-orbits among them becomes extremely large as the genus grows, although the paper notes that even monotonicity in $g$ is unknown.