Not with the type of reproduction that happened here. Explanation stolen from elsewhere since I'm too lazy to do it myself:
"The article claims only that the offspring is a perfect genetic "match" for the mom, not that it is identical to the mom since it also says the offspring has half the variability. What this means is the genetic test they did not pick up any polymorphisms not found in the mother. That's what they mean by "identical match".
Also parthogenesis does not create homozygotic offspring (although given enough generations it will), the immediate offspring is a result of a fusion event between 2 products of meiosis - the egg and one of the polar bodies. Thus the offspring will have a different genetic makeup to the mother. In particular half (on average) of the mother's heterozygous loci will become homozygous in the offspring. Thus the offspring has half the genetic variability."
So everywhere that had chromosomes AB has a 50% chance of still having that, and 25% chance each for AA and BB. If that baby has another one the same way then all of the AA and BB pairs will stay identical to the mother's ones but the AB pairs still have that 50% chance of changing. Repeat enough generations and you'll essentially have clones, with all the same AA and BB pairs and no AB pairs left.