The naive conjecture relating circular orders and circle actions

Let $G$ be a countable group. Write $\operatorname{CO}(G)$ for the space of circular orders on $G$, and let $\operatorname{Hom}(G,\operatorname{Homeo}_+(S^1))$ be the space of actions of $G$ on $S^1$ by orientation-preserving homeomorphisms. The naive conjecture. The space $\operatorname{CO}(G)$ has no isolated points if (or perhaps if and only if) $\operatorname{Hom}(G,\operatorname{Homeo}_+(S^1))$ is connected. The statement is motivated by the case $G=\mathbb{Z}^2$, for which both spaces have the corresponding properties; its general validity is not established here.