Invariant-metric conjecture for metrizable proper group actions

Let $G$ be a locally compact group and $X$ a metrizable proper $G$-space, where a proper $G$-space is a completely regular Hausdorff space with a continuous $G$-action satisfying Palais's small-neighborhood condition. A metric on $X$ is $G$-invariant if $d(gx,gy)=d(x,y)$ for all $g\in G$ and $x,y\in X$.

Invariant-metric conjecture. The topology of $X$ is metrizable by a $G$-invariant metric.

The source says that this conjecture is open and that, for metrizable $X$, it is equivalent to the paracompactness conjecture for orbit spaces of proper actions. It is an old problem attributed to R. Palais.