Gaussian wave packets spread as the time passes, which means that as we go backwards in time they get more localized. Therefore, Gaussian wave packets introduce less uncertainty in the position as the we go backwards in time.
Are there wave functions (not necessarily Gaussian) in Nature that don't behave like this, i.e that don't get localized as we go backwards in time? If so, I think this would mean that the uncertainty they introduce in position doesn't necessarily increase with time (or decrease as we go backwards in time) as it does in the case of a typical Gaussian wave packet.