Completely positive reduced dynamics with non-Markovian initial states

Consider an open quantum system S, interacting with its environment E, and also an ancillary Hilbert space R. The reduced dynamics of the system S  is given by a completely positive map, if the set S={ρSE}  , of possible initial states of the system-environment, can be written as a steered set from a tripartite Markov state ƮRSE. In this paper, we call such steered states  as Markovian states ρSE, and study a physical case, in which the reduced dynamics of the system can be completely positive, even when the initial states of the system-environment ρSE are non-Markovian.