Emergence of Noncontextuality under Quantum Darwinism

Quantum Darwinism proposes that the proliferation of redundant information plays a major role in the emergence of objectivity out of the quantum world. Is this kind of objectivity necessarily classical? We show that if one takes Spekkens' notion of noncontextuality as the notion of classicality and the approach of Brand\~{a}o, Piani and Horodecki to quantum Darwinism, the answer to the above question is `yes', if the environment encodes sufficiently well the proliferated information. Moreover, we propose a threshold on this encoding, above which one can unambiguously say that classical objectivity has emerged under quantum Darwinism.
Comment: Essentially the accepted version. Enhanced clarity, but main results stand the same. Addition of two new subsections: 1- a cutoff for classical objectivity 2-state spectrum broadcasting case (+ the relative subsections to the appendix). For the final version, please look at PRX Quantum. 8 pages, 2 figures in the main text + 7 pages, 1 figure in the Appendix. Comments are welcome!