The conjecture on almost classically Galois structures of non-abelian simple type

Let $N$ be a non-abelian finite simple group.

Almost-classical correspondence conjecture. The almost classically Galois Hopf--Galois structures of type $N$ are exactly the Hopf--Galois structures which admit a bijective correspondence.

This conjecture concerns the relationship between almost classically Galois Hopf--Galois structures and bijective correspondences for non-abelian finite simple types. The surrounding computations motivate the conjecture, but no resolution is given here.