Tucker's two-colour conjecture for connected locally finite graphs

Let $G$ be a connected, locally finite graph. The graph $G$ has infinite motion if every non-identity automorphism moves infinitely many vertices. A vertex colouring is asymmetric if the only automorphism preserving it is the identity. Tucker's conjecture. If $G$ has infinite motion, then there is an asymmetric colouring of $G$ with $2$ colours.

Despite numerous partial results, this conjecture remains wide open in general; it is a proposed infinite-graph analogue of the Motion Lemma for finite graphs.