I've been wary of running anything stateful on K8s for other reasons than the stable identity stuff. Historically, Docker's I/O has not been particularly robust — well, Docker has not been particularly robust. You also have to be very careful about resource limits and requests.
In particular, you have to ensure that the QoS class "Guaranteed" [1] kicks in, because the last thing you want is for Kubernetes to think that PostgreSQL is something that is happily rescheduled.
[1] https://github.com/kubernetes/community/blob/master/contribu...