The local-global realizability conjecture for inertia candidates in $\operatorname{GL}_2(\mathbb{F}_p)$

Let $p>2$ be a prime, let $G=\operatorname{GL}_2(\mathbb{F}_p)$, and let $(G,I,p)$ denote the realizability problem with inertia subgroup $I$. An inertia candidate is a subgroup $I$ satisfying the local conditions for an inertia subgroup. The local-global realizability conjecture. For every inertia candidate $I$, the triple $(G,I,p)$ is $\mathbb{Q}$-realizable; equivalently, the local-global principle is valid.

The conjecture is motivated by translating the realizability problem into the existence of modular forms associated to the required Galois representations. It remains unresolved in the supplied source.