Positive-measure prevalence of attracting Herman-ring cycles in real Desboves maps

Consider the parameter space of complex Desboves maps with real coefficients. A cycle of attracting Herman rings is a periodic collection of Herman rings whose return dynamics is attracting in the transverse direction. Positive-measure Herman-ring conjecture. There is a subset of positive measure in this parameter space consisting of maps with a cycle of attracting Herman rings. The claim is supported in the paper by comparison with the discussion of Herman rings and a numerical example, but is listed as an open conjecture.