  Modern quantum information theory provides new tools for investigating the
decoherence-induced "classicality" of open quantum systems. Recent observation
that almost all quantum states bear non-classical correlations [A. Ferraro {\it
et al}, Phys. Rev. A {\bf 81}, 052318 (2010)] distinguishes the zero-discord
classicality essentially as a pathology of the Markovian bipartite-systems
realm. Nevertheless, we formally construct such a classical model and its
variant that represents a matter-of-principle formal proof, i.e. a sufficient
condition for the, otherwise not obvious, existence of the Markovian
zero-discord classicality. A need for the more elaborate and more systematic
search for the alternative such models reveals we are still learning about the
very meaning of "classicality" in the realm of open quantum systems.
