The infinite motion conjecture for locally finite denumerable graphs

Let $\Gamma$ be a connected, locally finite, denumerable graph. Its automorphism group has infinite motion if it contains no automorphism with finite support, where the support of an automorphism is the set of vertices it moves. Infinite motion conjecture. If the automorphism group of $\Gamma$ has infinite motion, then $\Gamma$ has distinguishing number $2$. This conjecture extends the finite Motion Lemma, where sufficiently large motion compared with the group size implies distinguishing number $2$; the statement concerns the analogous claim for infinite automorphism groups and is presented here as an open conjecture.