Let $G$ be a locally compact, second countable group acting on a standard probability space $(X,\nu)$, and let $\nu$ be $G$-invariant. Let $G_x = \{g \in G\,:\, gx=x\}$ denote the stabilizer of $x \in X$. Assume that $G_x$ is cofinite in $G$ for $\nu$-almost every $x$. Equivalently, let there exist a $G$-invariant probability measure on almost every $G/G_x$. Does it follow that if $\nu$ is ergodic then it is supported on a $G$-orbit? That is, is the action then essentially transitive?

I know that this is true for discrete $G$: since orbits are finite, given a compact topological model for $X$, orbits are closed. Hence by Zimmer (1982) "Ergodic theory, group representations and rigidity" all invariant probability measures are supported on an orbit. It seems to also hold for Lie groups (Stuck-Zimmer). But does it hold in the more general, locally compact second countable setting?