The no-isolated-points conjecture for circular orders on free products

Let $G$ and $H$ be circularly orderable groups, and let $G*H$ denote their free product. Write $\operatorname{CO}(G*H)$ for the space of circular orders on $G*H$, with its natural topology. An isolated point is a circular order that is isolated in this topological space.

Free-product no-isolated-points conjecture. The space $\operatorname{CO}(G*H)$ either has no isolated points or is finite. In particular, if it is infinite, then $\operatorname{CO}(G*H)$ is a Cantor set.

This proposed generalization removes the infiniteness and minimality assumptions from the paper's theorem on non-isolated circular orders. The finite cyclic case illustrates why a finite alternative is necessary, and the general assertion remains open.