The main OIT conjecture for modular-tower monodromy

Let $O$ be a modular-tower orbit with geometric monodromy $G_O$, and for each $J'$ let $\mathcal G(\overline{\mathbf S}_{J'})$ denote the isomorphism class of the corresponding decomposition group. Main OIT conjecture. The geometric monodromy $G_O$ is an eventually $\ell$-Frattini sequence, and for each $J'$,

$$\mathcal G(\overline{\mathbf S}_{J'})\cap \mathcal G_O$$

is eventually $\ell$-Frattini. The conjecture connects the orbit-level geometric and arithmetic monodromy of modular towers with the expected Frattini structure.