When I drive my Prius at speeds over 42 mph, the engine runs all the time as we all know. If I watch the display on the dashboard, every now and then, the battery discharges. I believe this is why we can't tow the Prius without the ignition on. What is happening when driving is that the battery needs to discharge occasionally so it doesn't overcharge. In order for this to happen, I am guessing the ignition has to be on, and probably the engine also.
Ed Headington