One less simple idea that I meant to mention. It probably isn't the problem since the engine and everything else was off, but if you're under the hood might be worth trying. Unhook the ground wire from the battery and attach it to the ground point on the firewall behind the engine. This goes back to the second simple idea I mentioned. If the bracket goes into metal, the radio is probably grounded to the Jeep body at that point. Grounding to the battery creates a second ground path for the radio and it's the only direct line to the battery in the Jeep because everything else is also grounded to the body. All the power in the Jeep goes back to the battery, usually through the body to the firewall ground and then to the battery, but in some scenarios it will flow through the radio and use the radio's ground wire to the battery.