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Similar situation for me when my back up lights didn't work when car was placed in reverse. Mine's a 99 though, might be similar. Turns out it wasn't trans related, fuse, or bulbs. It was the wiring!
Check the bundle of wire that run down from the trunk lid, to hinge, and into the trunk. The wires connect to the reverse light, trunk light, and probably inner tail lights too. I didn't confirm this. I just cut and re-crimped every wire in the bundle.
Follow it into the trunk. There is a plastic housing that neatly hides the wires that is attached to the trunk lid hinge. Check for kinked wire(s) at the end of that plastic housing. That's where my wires were kinked. The wire tubing was still intact, but the wire inside was not making solid connection. It's like bending the paperclip back/forth until it breaks. Well my wires broke inside the wire tubing. I had to cut the kinked section and re-crimped the wires. I just bought wire looms and zipped tied it to the hinge. I didn't want to use that plastic housing again.
Toyota and Lexus Join Mille Miglia For The First Time
Slideshow: A five-car lineup spanning more than five decades of Toyota performance and engineering will tackle one of Italy's most celebrated automotive routes.