The theory of nonequilibrium (time-dependent) open dynamical systems is (naturally) much less developed than that of equilibrium closed systems. By nonequilibrium, we mean that the dynamical model itself varies in time. Unlike the setting of random dynamical systems, we do not assume any statistical knowledge of how the model evolves in time. By open, we mean that the phase space contains holes through which mass may escape. Studies of equilibrium (time-independent) open systems often focus on the existence of conditionally invariant measures and escape rates. Such conditionally invariant measures will not exist if the system is out of equilibrium. In this talk we discuss the concept of conditional memory loss for nonequilibrium open systems and we show that this type of memory loss occurs at an exponential rate for nonequilibrium open systems generated by one-dimensional piecewise-differentiable expanding Lasota-Yorke maps using convex cones and the Hilbert projective metric. We also explicitly estimate this rate.