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