the Jeffords AMX R with Rambleseat

Not the only Rambleseat AMX, which would cause me to think that there was a lot of desire around AMC to get a rumble seat back onto cars, which had been without them a really log time from a factory production car (T Birds had the "Birds Nest" aftermarket accessory) and I'll guess the last factory rumble seat production car was in the late 30s

this one is known as the Vignale AMX, worked on by the Torino Carrozzier for display at the 1966 New York Auto Show. It's now owned by Fred Phillips of Canada

I don't have info on this one I posted last summer  http://justacarguy.blogspot.com/2016/09/the-rumble-seat-amx.html and one is in the Crawford Auto and Aviation Museum

But it's not the same as the top Jeffords AMX R obviously. The JAMXR had side glass, and a hood with much larger fake airscoops

Safety concerns doomed the Rambleseat idea as AMC's designers revamped the Vignale AMX into the production AMX. In September of 1966, President Lyndon Johnson signed the National Traffic and Motor Vehicle Safety Act, which required all new vehicles sold in the United States to have, among other safety items, seat belts, shatter-resistant windshields and energy-absorbing steering columns. If the Act didn't ban rumble seats in letter, it certainly did so in spirit.

Jim Jeffords, meanwhile, arrived at AMC as manager and president of the Javelin Racing Team, Inc., to take the Javelin Trans-Am racing after his great success at racing Corvettes, including the famed Purple People Eater 1958 Corvette, and took SCCA's B Production championship in 1958 and 1959. He also bought the SR2 that Jerry Earl raced in the 56-57 season . One of 3 SR 2s

Jeffords felt that the Vignale AMX promised a Rambleseat AMX, and if AMC wanted to carry that good reception by fans, he'd have some made and sold through AMC dealerships, and his AMX R was even in the Dec 67 Rod and Custom.

AMC didn't seem to be a big fan of Jeffords though, replacing him with Roger Penske for the 1969 racing season, but they did keep the blacked out hood (later called the Shadow Mask option) and the ducktail spoiler on the rear edge of the roof

https://www.hemmings.com/magazine/mus/2009/11/Ramble-seat--On---1968-AMC-AMX-R/2291441.html
http://www.science.uwaterloo.ca/~afleming/jeffords_amx-r/