Topological Vaught conjecture for Polish group actions

Let $G$ be a Polish group and let $X$ be a Polish $G$-space, meaning that $G$ acts continuously on $X$. An orbit is a set of the form $G\cdot x$ for some $x\in X$; two points are equivalent when they lie in the same orbit. Topological Vaught conjecture. If $X$ has uncountably many orbits, then $X$ contains an uncountable closed set $C$ with no equivalent points: whenever $x,y\in C$ and $y=g\cdot x$ for some $g\in G$, one has $x=y$. This is a topological analogue of Vaught's conjecture concerning the number of models of a theory. The source provides references but no resolution, so the conjecture is treated as open here.