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Fair division is a subfield of multiagent systems that is concerned with object distribution. When objects are indivisible, the Maximin Share Guarantee (MMS) is a desirable fairness notion; however, it is not guaranteed to exist. While MMS allocations may not always exist, a relaxation of MMS is guaranteed to exist. We show that there exists a family of instances for which this relaxation fails to guarantee the MMS value for all but a small constant number of agents.