Ruling out critical Higgs inflation?

We consider critical Higgs inflation, namely, Higgs inflation with a rising inflection point at smaller field values than those of the plateau induced by the nonminimal coupling to gravity. It has been proposed that such a configuration is compatible with the present CMB observational constraints on inflation and also with primordial black hole production, accounting for the totality or a fraction of the observed dark matter. We study the model while taking into account the next-to-next-to-leading order (NNLO) corrections to the Higgs effective potential: such corrections are extremely important to reduce the theoretical error associated to the calculation. We find that, in the 3σ window for the relevant low energy parameters, which are the strong coupling and the Higgs mass (the top mass follows by requiring an inflection point), the potential at the inflection point is so large (and so is the Hubble constant during inflation) that the present bound on the tensor-to-scalar ratio is violated. The model is viable only when allowing the strong coupling to take its upper 3-4σ value. In our opinion, this tension shows that the model of critical Higgs inflation is likely to be not viable: neither inflation nor black holes as dark matter can be originated in this version of the model.