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Possibly not a rational answer (so possilbly not living up to the less wrong philosophy!) but given the assumption of an infinite plane I would think the probability is vanishingly small of returning to the original position and velocity.
Something would need to constrain the vectors taken to prevent any ball from taking off in some direction that could be described as "away from the group". Perhaps that could be understood be be on a path for which the the path of no other ball can possible intersect. At that point this ball can will never change it's current velosity and never return to it's oiginal position.
I cannot offer a proof that such a condition must eventually occur in your experimnt but my intuition is that it will. If so that vanishing small probablity that everyting return to some orginal state goes to zero.