From Jeff Traylor:
I found a guy on-line advertising a 1982 Frank Sinatra CONVERTIBLE. Did Chrysler ever actually make this model? or is this a custom job? Just wondering, I'd like the car if it is actually a Frank.
Here is the text of the ad as I found it on TraderOnline. Anyone interested in finding an Imperial should check this service once a week or so, they average 100 Imperials of all vintages for sale. Currently there are about 15 1981-1983 Imperials for sale.
1982 CHRYSLER IMPERIAL, Frank Sinatra SPECIAL, CONVERTIBLE, Leather Interior, 318 Fuel Injection, Automatic, Low Mileage, Very Good Condition, Inside and Out.
Recently occurred to me that it's probably a "roadster roof" fake convertible. The Mirada/Cordoba, which shares more than a few parts with the 81-83 Imperials, are the only cars I've ever seen where this roof treatment looks like the real deal.
From Chris Hoffman:
The 1981-83 Imperial never offered a factory convertible-look roof (nor an ordinary vinyl roof). You are probably thinking of the 1980-83 Dodge Mirada and Chrysler Cordoba, which used a completely different roof stamping from the Imperial coupe despite many shared underpinnings, some (very little) common sheetmetal, and a few pieces of shared glass. Actually, the Mirada's convertible look was relatively convincing, though it did make the car look kind of like a miniature Collector's Series Lincoln Mark V.
If you think about how the 81-83 Imperial's rear window and C-pillar extend down into the fender to give the bustle-trunk effect, the thought of a finishing a convertible top's seams here seems like an aesthetic challenge, to be polite, at best. Cutting the roof off an attractive coupe does not necessarily make for a more attractive convertible!
In July 2005, this car appeared for sale on eBay, with the following ad:
Car is in great condition and has no dents or dings in the paint. The interior is original and in mint condition. This car is the only  convertible ever made, it was originally bought in Florida then shortly after brought to Global Coach Convertible Experts in Orlando, Florida, for the convertible top installation. Convertible top is in perfect condition with no rips, tears, or any signs of age. The car has blue leather interior, factory a/c, hideaway front headlights, factory and not-converted original electronic fuel injection. The vehicle has it's original number matching 318 V8 engine with automatic transmission. Also equipped is the original heated outside mirrors. Vehicle is rust free and always been babied. Vehicle has 58800 original miles on it. Never driven in rain or snow.