Typically the VSR senses two voltages. The main battery and the aux battery. If it's not connected TO an aux battery it can't tell the drop between the two - which is uses to be intelligent. Some of these simpler VSRs I understand just sense drop to earth on the one battery so they should run most of the time with the engine. But the battery, once the engine is off, will drop pretty quickly to the point that it will deactivate the VSR. As you are pulling power from the battery, that higher float charge will deplete quite quickly I'd think. It will sense the draw and shut off. No I am not an electrical engineer, but I've had a few of these and that's how they seem to work.
I have never seen any of my chargers connect with the ignition on, but the engine not running. There has to be charge being sensed to the main battery before it will trigger.