Like Patrick, I had the privilege of spending a lot of time with the Carson car. I spent a particularly large amount of time with the car in 2008. In the months preceding the 2008 DCS, the show had very few concours entries registered (only 2 including Ken). As you know, Cliff (the then owner of the Carson car) was heavily invested in DCS along with Ken. For the sake of the show and having participants in the judging, Cliff was planning to enter the Carson car in the Concours judging. The car was still all original, never restored. I spent a full day going through the car with the Millennium Concours manual, doing a mock judging, and making a short list of any small items we could clean or correct between the car. The car did not need much at all. In the end, when the show rolled around there ended up be about half a dozen entries so Cliff decided not to have it judged, due to judging time constraints (so as not to take away from the other Concours cars). Regardless, I am 100% positive that it would have scored at least a bronze certification, if not a silver certification.
I'm not sure what your definition is of "not in the best shape," but to me, any car that will score in Concours is in pretty good shape. I'm sure the Sheas would validate this and plan to keep it that way.