In this paper, we consider a decentralized cooperative control policy proposed recently for steering multiple non-holonomic vehicles between assigned start and goal configurations while avoiding collisions. The policy is known to ensure safety (i.e., collision avoidance) for an arbitrarily large number of vehicles, if initial configurations satisfy certain conditions. The method is highly scalable, and effective solutions can be obtained for several tens of autonomous agents. On the other hand, the liveness properties of the policy, i.e. the capability of negotiating a solution in finite time, are not completely understood yet. In this paper, we introduce a condition on the final vehicle configurations, which we conjecture to be necessary and sufficient for guaranteeing liveness. We prove the necessity by a constructive method. Because of the overwhelming complexity of proving the sufficiency of such condition, we assess the correctness of the conjecture in probability through the analysis of the results of a large number of randomized experiments. [PDF]