There is a connector inside the liftgate, that provides current from the running lights circuit, to power the license plate lights. You may have a problem connection there, or with wiring running from the liftgate to the lower tail light wiring connection point.
If you consult the
wiring diagrams you will be able to see the circuit layout, as well as the routing of the harness that serves the liftgate and license plate lamps.
Those diagrams are for the 2004 year model, but there were not significant changes from what is presented in that particular manual during the run of the Gen1 Highlanders.