I don’t know your switching system, but if both the left and right cameras are working OK with turn signal triggers, can you confirm that the rear camera is connected properly and works OK by temporarily plugging that rear camera feed into a known good side camera input? If the rear camera and feed cable work OK in that temporary scenario, then perhaps the transmission signal in reverse gear to switcher, is not actually doing its thing. I’ll defer to those that actually know your system though.
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