Paracompactness conjecture for orbit spaces of proper group actions

Let $G$ be a locally compact group, and let $X$ be a completely regular Hausdorff space with a continuous action of $G$. The action is proper if every point has a neighborhood $V$ such that, for every point of $X$, some neighborhood $U$ satisfies that $\{g\in G\mid gU\cap V\neq\emptyset\}$ has compact closure in $G$. The orbit space $X/G$ is the quotient of $X$ by the $G$-action, and a space is paracompact if every open cover has a locally finite open refinement.

Paracompactness conjecture. If $X$ is a paracompact proper $G$-space, then the orbit space $X/G$ is paracompact.

This is described as a major open problem in the theory of proper actions. It remains open even when $X$ is metrizable, and concerns the transfer of paracompactness from a proper $G$-space to its orbit space.