The higher-dimensional double midset conjecture

Let $X$ be a non-degenerate compact metric space, and suppose that every midset of $X$ is homeomorphic to the $(n-1)$-sphere $S^{n-1}$. Higher-dimensional double midset conjecture. Then $X$ is homeomorphic to the $n$-sphere $S^n$. This extends the double midset conjecture to the setting in which all midsets have the topology of spheres; the source does not provide a resolution, so the conjecture remains open.