I don’t think they do. They seem to treat the previous output as the current input, and as a general rule you can’t backpropagate your inputs. So unless they say very clearly that they do backpropagate through the previous outputs, then they don’t.
That said, if you are still in doubt, why don’t you email the authors and ask them? If you do, try to pose your question as clearly as possible because if they don’t clearly understand the question then they might not find the time to reply.