Just to provide you with some information we figured out during the last weeks.
The mentioned error is triggered by the readiness probe on the path: /state
When we changed it to /health like for liveness probe all zero pods work like expected without this error message.
We have also found a pull request from year 2020 where the issue is already mentioned. However the path: /state is still set in the value.yaml file for helm charts provided on GitHub (which caused the mentioned issue in my original post).
Are there any reasons to do the readiness probe for zeros on the path: /state ?