The infinite motion conjecture for graphs

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

Infinite motion conjecture for graphs. If $\operatorname{Aut}(G)$ has infinite motion, then

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

Automorphism groups of connected, locally finite graphs are closed and subdegree-finite, so this conjecture follows from the corresponding permutation-group conjecture. It is known for several growth ranges, including super-linear subquadratic growth and superpolynomial subexponential growth, but remains open for general locally finite connected graphs.