It’s easy to see that Paxos does have a failure mode. When two proposers are active at the same time, they may ‘duel’ for highest proposal number by alternately issuing proposals that ‘one-up’ the previous proposal. Until this situation is resolved, and a single leader is agreed upon, it’s possible that Paxos may not terminate. This violates a liveness property. However, the likelihood is that eventually Paxos will return to a correct execution once the network settles down and the two proposers observe each other and agree to let one go first (note that this isn’t quite the same as solving consensus: one proposer simply has to back off for sufficiently long to let the other proposer get its proposal committed).