This is not the only assumption. There is also the assumption of "factorizably HSD probabilistic theory". Look at the motivation of this assumption immediately after Proposition 1. Among other things, it looks like this motivation is only good for finite-dimensional systems.The only other assumption is that two level systems exist, i.e. qubits. That seems to be obviously true right?
Another thing. Is existence of two-level systems so obvious? I guess this depends on exact definitions, because all realistic systems are multilevel systems.