A friend of mine who is into these things, and has read the article, has responded privately:

"I think it's quite possible that we reincarnate with our animals. They're part of our karmic fabric and script. The way I understand it is that animals have group souls. The less evolved the animal the larger the numbers which make up the group soul. But animals in close contact with humans, especially cats, dogs and horses will have much smaller soul groups (maybe less than ten). They will therefore be the first ones to individualise and graduate to the human kingdom in the next Round (Round 5). At some point during the current Round the door was closed for further animals to humanise in this cycle".