I always unplug our trailer and switch the battery disconnect to OFF for the Winter. Then in the Spring, check the level in the batteries, plug the trailer in and turn the battery disconnect to ON. Never had to replace the batteries in nine years.
We leave ours plugged in, batteries in, and check them periodically. We have four deep cycle due to our solar system, so leaving them in makes more sense, but again we check them periodically
2011 Bighorn 3055RL, 2011 F350 DRW 6.7L 4x4 Diesel Lariat and Hensley TrailerSaver BD3, 1992 Jeep ZJ and 1978 Coleman Concord Pop-Up for remote camping Dave & Renee plus (Champ, Molly, Paris, Missy, and Maggie in spirit), Mica, Mabel, and Melton
Ours gets shut down and batteries turned off. I will check the batteries about once a month, and if they are not still fully charged, I will plug the rig in, turn on the batteries and have the converter charge the batteries for about 24 hours.
Many leave theirs plugged in all year, even in extreme cold. I unplug, disconnect and remove my battery before the snow flies. That's just me, neighbor right across the street leaves his FW connected, and battery charging all year.
In any case, batteries should be checked, in or out of the unit.