Small loops condition for the constructed non-Euclidean examples

Determine whether the limit continuum \(K\subset S^n\) in the constructed contractible, non-Euclidean locally conformally flat manifolds with positive scalar curvature satisfies the small loops condition.

Background

The paper constructs, for every dimension $n\geq4$, a nondegenerate continuum $K\subset S^n$ whose complement is contractible but not simply connected at infinity, and equips the complement with complete locally conformally flat metrics of positive scalar curvature.

These examples demonstrate that the two topological hypotheses in the Euclidean-rigidity theorem cannot be omitted simultaneously. However, the construction does not determine whether the conformal boundary continuum satisfies the small loops condition, so it does not show whether either hypothesis can be omitted while retaining the other.