The infinite motion conjecture for permutation groups

Let $A$ be a permutation group acting on a countably infinite set $X$. The group is subdegree-finite when every orbit of a point stabilizer is finite, and it has infinite motion when every nonidentity element moves infinitely many points.

Infinite motion conjecture for permutation groups. If $A$ is closed and subdegree-finite with infinite motion, then

$$\operatorname{D}(A,X)=2.$$

This extends the finite-motion Motion Lemma to closed, possibly uncountable permutation groups. The conjecture is motivated by counterexamples among closed groups that are not subdegree-finite, but remains open in the stated generality.