  We study a recent conjecture about the behavior of the quantum relative
entropy compared to the relative entropy of entanglement in open bipartite
systems. The conjecture states that, under a dissipative time-evolution, the
positive rate of change of the relative entropy will always be larger than that
of the relative entropy of entanglement. After explicitly solving a two-qubit
master equation of Lindblad type with separable and entangled stationary
states, we show that the conjecture can be violated for initial states with an
entangled asymptotic state, while it appears to be confirmed when the
asymptotic states are separable.
