Resolving this question would provide a template for understanding mixing at criticality more broadly, a regime largely absent from the rigorous cutoff literature. It would clarify whether algorithmic choices like scan order matter for the sharpness of convergence near phase transitions, which is directly relevant to the practical efficiency of systematic scan MCMC in statistical physics and Bayesian computation. It would also likely require new techniques connecting spectral theory of non-reversible chains, renormalization group ideas near criticality, and precise large deviation estimates, advances that would have independent value across probability theory and the theory of algorithms.