Asymmetric colouring of locally compact permutation groups

Let $G\le \mathrm{S}\mathrm{y}\mathrm{m}(X)$ for a countable set $X$. Call a colouring of $X$ asymmetric, if the identity is the only element of $G$ which preserves all colours. The motion (also called minimal degree) of $G$ is the minimal number of elements moved by an element $g\in G\setminus \{\mathrm{i}\mathrm{d}\}$. We show that every locally compact, closed permutation group with infinite motion admits an asymmetric $2$-colouring. This generalises a recent result by Babai and confirms a conjecture by Imrich, Smith, Tucker, and Watkins from 2015.