Update. It seems that it's not just re-displaying the last 20 when a new post comes in, it's doing it even when there hasn't been anything new since the last update.
Therefore, whenever a mobile user goes online, he may want to check all replies tweets that were sent to him while he was offline. "Retrieve Tweets History AfterLogin" is for this requirement. By enabling it, this plugin will save a user's state when he goes offline. It can restore the state when the user goes online next time, thus retrieving replies exactly since the last reply he received.