Ok, reading that paper now.
My current understanding is: if an attempt is made to lock a resource on some shard with some lock ID, and this attempt fails because the resource is already locked, then this issues a log that can be used to unlock any resources in other shards that were locked in the same lock ID. If a single shard receives notifications of logs from all shards, then it knows that there are no shards that could have issued a log that would repeal any of the other locks, and so it can safely publish a confirm message.