The motion conjecture for graphs

Let $G$ be an infinite graph of arbitrary cardinality. Write $\operatorname{m}(G)$ for its motion, $\operatorname{Aut}(G)$ for its automorphism group, and $\operatorname{D}(G)$ for its distinguishing number.

Motion conjecture for graphs. The inequality

$$2^{\operatorname{m}(G)}\geq |\operatorname{Aut}(G)|$$

implies

$$\operatorname{D}(G)=2.$$

This is the infinite-cardinality form of the Motion Conjecture for Graphs and would extend the Motion Lemma beyond finite automorphism groups. The source presents it as a conjecture and gives no resolution in the stated generality.