Which groups can occur as 2-knot groups with non-trivial normal bundles

Characterize the finitely presented groups that can arise as 2-knot groups, i.e., fundamental groups of complements of smoothly embedded 2-spheres, in the case where the 2-spheres have non-trivial normal bundles in closed 4-manifolds, without imposing any exoticness requirement.

Background

The construction in the paper yields exotic 2-links with trivial normal bundles and realizes any finitely presented group as the 2-link group in an appropriate simply connected 4-manifold. However, the situation for 2-spheres with non-trivial normal bundles is subtler.

Gauge-theoretic constraints (e.g., results of Kronheimer–Mrowka and more recent work) indicate restrictions in this setting, but a complete understanding of which groups can occur as complements of such 2-knots remains incomplete.