Tucker's infinite motion conjecture for locally finite graphs

Let $X$ be a connected, locally finite graph. The motion of $X$ is

$$m(X)=\inf\{m(f)\mid f\in\operatorname{Aut}(X),\ f\neq\operatorname{id}\},$$

where $m(f)$ is the cardinality of the set of vertices not fixed by $f$. The graph $X$ has infinite motion when $m(X)$ is infinite.

Infinite motion conjecture. If $X$ is a connected, locally finite graph with infinite motion, then

$$D(X)\leq 2.$$

This conjecture proposes that every connected, locally finite graph whose nontrivial automorphisms move infinitely many vertices admits a distinguishing vertex coloring with two colors. The paper proves the conjecture under additional hypotheses, but the general statement is presented as an open problem.