The no-elliptic-ramification conjecture for Modular Tower components

Assume the Main Conjecture hypotheses, and let $\bar O_k$ be any component at level $k$. Elliptic ramification from $\bar O_k$ to a component at level $k+1$ means the ramification phenomenon in which an isomorphism between the associated covers at level $k$ does not lift to an isomorphism at level $k+1$. No-elliptic-ramification conjecture. For $k$ sufficiently large, there is no elliptic ramification above any component at level $k$. This is a working consequence expected under the Main Conjecture hypotheses; the source gives no resolution.