The double midset conjecture for continua

A continuum is a non-degenerate connected compact metric space. For a metric space, the double midset property (DMP) means that every pair of distinct points has a midset consisting of exactly two points. Double midset conjecture. A continuum having the DMP must be a simple closed curve. The conjecture remains open, although it is known under several additional hypotheses, including the existence of a subcontinuum with no cut points, continuity of the midset function, or containment in the Euclidean plane.