In the theory of p-adic modular forms there is a certain construction called the Coleman-Mazur eigencurve. The spectral halo conjecture roughly states that if you remove a closed subdisc of the weight space, the eigencurve is an infinite disjoint union of finite flat covers of what remains of the weight space.

For a person who does not intrinsically care about the eigencurve, what interest does the halo conjecture pose? What corollaries does it have that do not directly involve the eigencurve?