Fundamental groups of homology 4-spheres and complements of contractible submanifolds

Fix a prime $p$, and for a group or space $X$ let $\beta_i^p(X)$ denote the dimension of $H_i(X,\mathbf{Z}_p)$. A homology 4-sphere is a 4-manifold with the homology of the 4-sphere, and a contractible submanifold is a submanifold of the 4-sphere that is contractible. Homology-sphere complement conjecture. A group is the fundamental group of a homology 4-sphere if and only if it is the fundamental group of the complement of a contractible submanifold of the 4-sphere. This is an open problem because there is no characterization of the fundamental groups of homology 4-spheres, and it is also unknown whether such a characterization must depend on the smooth or topological category.