The paper explains this:

The “weakness” of the Liveness property corresponds to the fact that we do not guarantee (though it is still hard for an attacker to prevent) a resolution in case conflicting transactions were published soon one after the other. Contrast this to traditional consensus protocols, where all conflicts are required to be decided in finite time, a property usually referred to as Liveness. Observe, however, that an honest user of the system will never publish conflicting transactions,and will transfer money only after he robustly accepted the original funds (the inputs) himself; payments of honest users are thus guaranteed to meet the conditions formalized in Weak Liveness, and to be robustly accepted. On the other hand, an attacker trying to defraud must keep his attack secret before publishing the conflict, until the victim robustly accepts; but then the victim is guaranteed that w.h.p. his transaction will not be reversed. Therefore, these two properties together ensure that payments of honest users will be robustly accepted in constant expected time, and that they remain robustly accepted forever, w.h.p.