The way I understand it, the author is not saying that the writers need to read the value but that they should check the latest state of the cluster's write quorum (i.e. can the writer see all 'w' nodes) and should hold off writing if it is not met. Otherwise, you can end up reading stale data of a write that was never deemed successful in the first place, even if the subsequent read was a 'read-repair', thereby violating linearizability.
In the quoted example (n=3, w=3, r=2), say the writer was only connected to Replica-1 (and not all 3 of them) and yet initiated a write of x=1, which of course went on to be considered a failure (since replicas 2 and 3 did not get updated). Now if the Reader A and Reader B did the read-repair as per the diagram, using latest-write-wins, all 3 replicas will have x=1. This is incorrect because readers must have gotten x=0 since the earlier write was deemed a failure.
Note that the readers A and B could have begun after we get the response for the write and we would still end up in the same situation. It is also possible that you can end up in this state even if the writer can talk to all 3 replicas but say the write failed on 2 of them due to disk errors. So checking the quorum before write is only a partial requirement.
In summary, any leaderless replication just relying on quorum based consistency is not 100% linearizable.
