J or SJ?

Technically speaking, there was really no such thing as a 'Duesenberg Model SJ," just as there are no Duesenberg SSJ's or Duesenberg Model A's (they are really just extra short-wheelbase J's and Straight Eights, respectively). A Duesenberg Model J that was supercharged was simply known by the factory as a 'supercharged Model J.' Nothing more, really. Even if the car was not among the 36 cars that were originally equipped with a supercharger, it is still a 'supercharged J' because, well, that is what it is. The SJ deisgnation has just come into the venacular as quick way to say 'supercharged J,' I guess.

I'm compiling a list of Duesenbergs and I have a question: When you add a supercharger to an unsupercharged engine, is the car known as an "SJ" or is it still a "J?"
Just to be clear, I mean adding a supercharger to an engine THAT HAS NEVER BEEN SUPERCHARGED BEFORE--in other words, converting a car from J to SJ specification.