Uniqueness of the k-sector for disjoint closed sets

Let $P$ and $Q$ be two disjoint nonempty closed sets in Euclidean space, and let a $k$-sector be the geometric object associated with $P$ and $Q$ that separates their distance-based regions. Uniqueness conjecture. The $k$-sector of any two disjoint nonempty closed sets in Euclidean space is unique. The authors report numerical experiments in the Euclidean plane whose lower and upper estimates did not differ significantly, motivating this conjecture; no proof or resolution is given, and the computational complexity of the approximation methods is left as a future research problem.