The abstract-index admissibility conjecture

Let $p \in \mathbb{P} \cup \{0\}$, and let $(\mathcal{G},\mathcal{H},\theta)$ be a $p$-embedding of abstract indices. The abstract-index admissibility conjecture. Then $(\mathcal{G},\mathcal{H},\theta)$ is $k$-admissible for some field $k$ with characteristic $p$.

Here, $k$-admissibility means that there exist connected reductive algebraic $k$-groups $H \subset G$ whose embedding of indices is isomorphic to $(\mathcal{G},\mathcal{H},\theta)$. The conjecture asserts that the definition of a $p$-embedding is sufficient for realizability over some field of characteristic $p$, extending the fact that every abstract index is admissible over some field.