I don't want to scare anybody because most of the time leaving the charger is fine. But if a battery in the loop goes bad (batteries banked together) you might risk a fire. It happened to me and luckily I caught it before the battery melted down, blew, or caught fire. The battery was so hot I could not touch it. My inverter/multi stage charger set to the charge mode puts out up to 100 amps. A battery temp sensor is on one battery but I was charging 4 (2 house-2 starting) and one of the starting batteries failed. Acid spewed all over and the next morning the smell tipped me off something was going on. The problem was I was getting weird reading off my gauges well in advance
I now know what to look for. I was seeing over 40 amps charge when after a few hours the amp gauge should have dropped down to a few amps. The next morning it was pumping out over 70 amps! Even smart chargers can be fooled into thinking a battery needs a charge and a bad battery can make the charger react this way. There are a host of ways to deal with it but basically you just need to monitor the charging when possible. The issue is rare but I'm overly cautious these days.
BTW some inverters can be set up to supply a maintenance charge but when set that way if a battery goes bad & is is banked with another that's not isolated from the other in some way, the bad battery will take the other or others down to nothing in a hurry as the charger can't keep up. Just another reason to always monitor the charge. AGM batteries can stay fully charged in normal weather for a month or longer provided there is no load applied. I always plug in while on the carport but am set to maintenance.