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The determination of the stability of the long-lived consensus problem is afundamental open problem in distributed systems. We concentrate on thememoryless binary case with geodesic paths. We offer a conjecture on thestability in this case, exhibit two classes of colourings which attain thisconjectured bound, and improve the known lower bounds for all colourings. Wealso introduce a related parameter, which measures the stability only forcertain geodesics, and for which we also prove lower bounds.

Geodesic stability for memoryless binary long-lived consensus