Last winter I left the two Trojan T-105s in the TT and the converter running.
During the winter, the converter failed and the batteries ran down (propane detector, etc). The coldest it went down to was a few degrees below zero. That was enough to freeze the electrolyte and crack the cases on both batteries (since discharged batteries freeze at a lower temp than charged batteries).
If I left the batteries in the TT with the converter running again, I would leave a small light on in the TT so I could see there was 12V power just by looking out the window at night.
Steve