Interestingly, it looks like Cortex's support for local storage and object stores comes from using Thanos's storage engine. So once it's production ready in Thanos it will probably be production-ready in Cortex shortly thereafter.
https://cortexmetrics.io/docs/operations/blocks-storage/
I think for Cortex your safest storage options now are Bigtable, DynamoDB, or Cassandra.