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This paper deals with computation of the capacity region of the discrete memoryless multiple-access channel (MAC), which is equivalent to solving a difficult nonconvex optimization problem. In the literature, it is claimed that for elementary MACs, i.e., MACs for which the size of output alphabet is greater or equal to the sizes of all input alphabets, the Karush-Kuhn-Tucker conditions provide a necessary and sufficient condition for sum-rate optimality. In this paper, we demonstrate that this claim does not hold, even for two-user channels with binary input and binary output alphabets. Consequently, the capacity computation problem for the discrete MAC remains an interesting and mostly unsolved problem.