Yes, but ever so slowly. With two batteries you should be okay for a day's travel. Remember, no one will be opening the fridge except for lunch, during they day, so the compressor won't be firing up and drawing down the batteries. Just my personal experience.
2015 Black F-350 Platinum Diesel Dually Crew Cab
2019 Riverstone Legacy 38MB (Full Paint/All Options)
2015 Voltage 3990 (Full Paint, All Options) Sold 6/18
2005 Harley Ultra Classic w/too many miles