Broere–Pilśniak's infinite edge motion conjecture

Let $G$ be a countable, connected graph. An automorphism moves infinitely many edges if it does not fix infinitely many edges of $G$. A $2$-edge colouring is distinguishing if the only automorphism preserving it is the identity. Infinite edge motion conjecture. If every non-trivial automorphism of $G$ moves infinitely many edges, then there is a distinguishing $2$-edge colouring. The conjecture is the edge-colouring analogue of Tucker's infinite motion conjecture and does not require local finiteness; it is proved in this paper.