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The maximum known asymptotic merit factor for binary sequences has been stuck at a value of 6 since the 1980s. Several authors have suggested that this value cannot be improved. In this paper, we construct an infinite family of binary sequences whose asymptotic merit factor we conjecture to be greater than 6.34. We present what we believe to be compelling evidence in support of this conjecture. The numerical experimentation that led to this construction is a significant part of the story.