The broad approach they're describing was described by Adya in 2019 as a LInK store. Linked In-memory Key-value stores allow for application objects supporting rich operations to be sharded across a cluster of instances.
I would tend to approach this by implementing a stateful application using Akka (disclaimer: I am at this writing employed by Lightbend, which employs the majority of the developers of Akka and offers support and consulting services to clients using Akka; as my SO history indicates, I would have the same approach even multiple years before I was employed by Lightbend) along these lines.
Akka Cluster to allow a set of JVMs running an application to form a cluster in a peer-to-peer manner and manage/track changes in the membership (including detecting instances which have crashed or are isolated by a network partition)
Akka Cluster Sharding to allow stateful objects keyed by ID to be distributed approximately evenly across a cluster and rebalanced in response to membership changes
These stateful objects are implemented as actors: they can update their state in response to messages and (since they process messages one at a time) without needing elaborate synchronization.
Cluster sharding implies that the actor responsible for an ID might exist on different instances, so that implies some persistence of the state of the zone outside of the cluster. For simplicity*, when an actor responsible for a given zone starts, it initializes itself from datastore (could be S3, could be Dynamo or Cassandra or whatever): after this its state is in memory so reads can be served directly from the actor's state instead of going to an underlying datastore.
By directing all writes through cluster sharding, the in-memory representation is, by definition, kept in sync with the writes. To some extent, we can say that the application is the cache: the backing datastore only exists to allow the cache to survive operational issues (and because it's only in response to issues of that sort that the datastore needs to be read, we can optimize the data store for writes vs. reads).
Cluster sharding relies on a conflict-free replicated data type (CRDT) to broadcast changes in the shard allocation to the nodes of the cluster. This allows, for instance, any instance to handle an HTTP request for any shard: it simply forwards a representation of the important parts of the request as a message to the shard which will distribute it to the correct actor.
From Kubernetes' perspective, the instances are stateless: no StatefulSet or similar is needed. The pods can query the Kubernetes API to find the other pods and attempt to join the cluster.
*: I have a fairly strong prior that event sourcing would be a better persistence approach, but I'll set that aside for now.