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I searched the forums and found many with similar power antenna problems. I would like to check the relay and I see many references to where the relay is located for the coupe but not the convertible.
As the previous poster mentioned check you have power, the antenna will not work if the grease has all dried up.
Remove the antenna and drill the rivits holding the plastic case together, remove the clips and pull it appart.
If the antenna cable has broken replace that, if the grease is gone grease the gear and the wheel assembly the antenna cable goes into so it doesnt jam on the antenna plastic case.
Reassemble and replace the rivits with a bolt and nut.
My 87 vert antenna wouldn't shut off a couple months ago when I parked the car. I reached behind the drivers seat in the rear compartment and pulled the carpet back and found two electrical connectors. I simply follow the wires with my hands to the antenna and unplugged that connector and the it cut power to the antenna.
I suspect my antenna motor gear is stripped as the antenna wasn't going up or down. I just put the antenna up about 6 inches and have left it as is....
The connectors are not hard to find, but they are very hard to actually see. Alot of hand work and judgement by feel is what I did it.
I found it, it was on the drivers side attached to the underside of the back deck, needed the tonneau cover open and to keep the back of the convertible top up to see it. I will have to get my 12 year old to crawl in there and get it out for me.