Actually, I think things might be even better than this: in semiclassical quantum gravity in asymptotically de Sitter spacetime, your quantum state is necessarily mixed (due to tracing over things outside the cosmological horizon, which is how you get Unruh radiation). So, there are no quantum Poincare recurrences. If you plug a stationary mixed state into the FCR interpretation, all the observables become completely frozen in time. If you’re just converging towards a stationary mixed state, I expect the observables to converge towards becoming frozen.
Actually, I think things might be even better than this: in semiclassical quantum gravity in asymptotically de Sitter spacetime, your quantum state is necessarily mixed (due to tracing over things outside the cosmological horizon, which is how you get Unruh radiation). So, there are no quantum Poincare recurrences. If you plug a stationary mixed state into the FCR interpretation, all the observables become completely frozen in time. If you’re just converging towards a stationary mixed state, I expect the observables to converge towards becoming frozen.