Kaiser-Frazer did some work along the lines of getting a convertible into production but lack of money (and lack of interest in terms of car sales after the 1951 model year) killed the program.
The most interesting concept that I saw appeared in a 1951 or early 1952 issue of MOTOR TREND magazine. MT ran a contest for readers dealing with what people thought a HJ convertible might be like and offered a cash prize in an amount the magazine thought a regular 2-door sedan might be converted for. The winner based his concept on the Nash Rambler platform in that the center section of the roof was removed and the window frames for doors and rear seat passengers remained. Tracks were set up along the inside of the frames and the top would be manually pulled up and would lock near the center peak of the windshield area (like I say, use the set up of the Nash Rambler's top in 1951 to see how the mechanics would work).
Unfortunately the only MT issue I have at present referring to the contest mentions the contest would end soon. All too many years ago though, I did see the issue presenting the winning idea.