Computing a square that simultaneously bisects three point sets

Determine how fast one can compute, given three point sets in the plane, a square that simultaneously bisects all three point sets.

Background

The paper's existence theorem shows that any three mass distributions in the plane, and hence suitable point-set measures, admit a simultaneously bisecting square. However, the topological proofs do not directly yield algorithms.

The authors identify the computational complexity of finding such a square as an unresolved algorithmic question. They note that a degree-based proof may sometimes be adapted into an efficient algorithm, while also explaining that a direct continuation method is complicated by the fact that three points generally lie on the boundary of infinitely many squares.