The index-three conjecture for Kontsevich–Zorich monodromy of origamis in $\mathcal{B}_d$

Let $\mathcal{O}$ be an origami of degree $d$ and genus $2$, let $\Gamma\subseteq\operatorname{SL}_2(\mathbb{Z})$ be its Kontsevich–Zorich monodromy, and let $\mathcal{B}_d$ denote the $\operatorname{SL}_2(\mathbb{Z})$-orbit of primitive origamis with odd degree distinguished by the HLK-invariant. Index-three conjecture. In the setting of the main theorem, if $d$ is odd and $\mathcal{O}$ lies in $\mathcal{B}_d$, then

$$[\operatorname{SL}_2(\mathbb{Z}):\Gamma]=3.$$

The paper proves the corresponding index bound in the odd case and establishes index one for the orbit $\mathcal{A}_d$; this conjecture completes the result for odd-degree primitive origamis by determining the remaining orbit.