As I recall, that system's security claims assumes that the users have a synchronous network-- every message is reliably delivered in order to all users.  This is a rather nonphysical assumption, the only way we know how to construct such a thing is to use some kind of consensus system. Which would make the name "Ouroboros" rather fitting. :P

If you did actually have a synchronous network, you wouldn't need any kind of POW/POS whatever, you'd just have people send their transactions and the first one sent wins. As a result under that assumption you could prove virtually any kind of system secure, even ones with absolutely no security in the real world.