I don't have an LR2, so I'm just thinking out loud. I would think there's a switch or sensor that is activated when reverse is selected. This would tell the backup camera to turn on. Maybe that has taken a crap.
You could pull the camera and put a voltmeter to the wires. Obviously the vehicle will have to be in reverse to see if the camera is getting power, so use good judgement. Set the parking brake or better yet, have someone sit in the seat with the parking brake set and their foot on the brake pedal. If you're getting power to the camera, then it is probably the camera that has gone bad. If you're not getting power to the camera, then it could be anything in between, including a switch, wiring, relay, ECU, or whatever else may be used to operate the camera.
Colin