In Section 4.2, the paper proves that in the common paradigm atomicity (as defined in Definition 4.2) is impossible. Then it says “Consequently, payment channel protocols adhering to this design are fundamentally unable to satisfy the essential property of balance security.”. I think that this statement is not correct because it assumes that a state update changes the balances. A protocol in the common paradigm in which state updates do not change the balances can achieve balance security.
Actually, Lightning can be seen as such a protocol in which state updates do not change balances: During a state update in Lightning, HTLCs are added and removed but balances are not changed. Balances are changed in Lightning only when an HTLC is fulfilled. Therefore, Lightning can achieve balance security even in the common paradigm.