Half-distance separator by a two-dimensional manifold in R^3

Determine whether, for any two disjoint continua A,B ⊂ R^3 (in particular, for disjoint simply connected continua), there exists a compact connected two-dimensional manifold C ⊂ R^3 such that A and B lie in different components of R^3 \ C and the Euclidean distance satisfies ρ(C, A ∪ B) = ρ(A,B)/2.

Background

In the plane, under suitable chainedness hypotheses, the authors prove that two separated sets A and B can be separated by a simple closed curve C whose distance to A ∪ B equals ρ(A,B)/2, and C is equidistant from A and B. Extending this phenomenon to three dimensions admits two notions of separators: spherical (homeomorphic to S^2) and topological (compact connected two-dimensional manifolds).

The paper shows counterexamples ruling out the spherical analogue and gives examples indicating that some topological manifold separators at half-distance can exist. However, the general validity of the topological analogue remains unresolved. The authors pose an explicit question asking whether a compact connected two-dimensional manifold achieving the half-distance separation exists for all pairs of disjoint (possibly simply connected) continua in R^3.