What They Saw
They used A. vinelandii ATCC12837 and knocked out the nifY gene (whose product makes the central Mo cofactor) with a kanamycin resistance insertion, then grew it with or without nitrogen or kanamycin. It shouldn't be able to fix nitrogen with Mo present, though maybe with Mo absent.
Looking at the methods, I couldn't find one of the restriction sites they claim to have used, but it's possible that it was present in their strain and not in mine. All the others seem to be there.
Actually, they transformed cells, plated them out to obtain single colonies on non-selective plates, and then screened these colonies for kanamycin resistance. The idea was that if each cell had multiple copies of its chromosome, only some of the offspring of a resistant cell would have kan resistance, because some would inherit the resistant version of the genome and others wouldn't.
But when they grew more colonies from originally resistant colonies and tested their resistance, all of them were resistant that came from resistant colonies. So once the kan-resistant phenotype was present in a cell, it got passed to all offspring; no evidence of polyploidy.
This study is lacking in some ways: I would've liked to see confirmation of the locus of insertion by sequencing, to make sure it's in the right place. And it would be interesting to see what happened if they selected for nitrogen fixation, as in 254. So the question isn't quite settled.