oauth2-proxy authentication calls slow on kubernetes cluster with auth annotations for nginx ingress

Given that it's unlikely that someone comes up with the why this happens, I'll answer my workaround.

A fix I found was to set the annotation to the following

  nginx.ingress.kubernetes.io/auth-url: "http://oauth2.infra-system.svc.cluster.local/oauth2/auth"  nginx.ingress.kubernetes.io/auth-signin: "https://oauth2.domain.com/oauth2/start?rd=/redirect/$http_host$escaped_request_uri"

The auth-url is what the ingress queries with the cookie of the user. Hence, a local DNS of the oauth2 service is the same as the external dns name, but without the SSL communication and since it's DNS, it's permanent (while the cluster IP is not)

In my opinion you observe the increased latency in response time in case of:
nginx.ingress.kubernetes.io/auth-url: "https://oauth2.${var.hosted_zone}/oauth2/auth"
settingdue to the fact, that auth server URL resolves to the external service (in this case VIP of Load Balancer seating in front of Ingress Controller).

Practically it means, that you go out with the traffic outside of the cluster (so called hairpin mode), and goes back via External IP of Ingress that routes to internal ClusterIP Service (which adds extra hops), instead going directly with ClusterIP/Service DNS name (you stay within Kubernetes cluster):

nginx.ingress.kubernetes.io/auth-url: "http://oauth2.infra-system.svc.cluster.local/oauth2/auth"