Deployment on AWS EKS - any chown-operation on persistent storage is not permitted

I deployed the chart on my aws eks cluster.
When I don't use persistent storage, all is fine.
But when I provide an EFS as persistent storage the nextcloud image is not able to perform any chown-operations.
Talked to AWS Support, but they couldn't help me.

How to reproduce:

helm install nextcloud w/ persistent storage on aws eks cluster.
helm upgrade --install -f value.yaml nextcloud nextcloud/nextcloud

logs of the nextcloud pod are this *1000:

rsync: [receiver] chown "/var/www/html/3rdparty/aws/aws-sdk-php/src/data/cloudfront/2019-03-26/.smoke.json.php.OCBd81" failed: Operation not permitted (1)
rsync: [receiver] chown "/var/www/html/3rdparty/aws/aws-sdk-php/src/data/cloudfront/2019-03-26/.waiters-1.json.php.qjGStZ" failed: Operation not permitted (1)
rsync: [receiver] chown "/var/www/html/3rdparty/aws/aws-sdk-php/src/data/cloudfront/2019-03-26/.waiters-2.json.php.e3lKv1" failed: Operation not permitted (1)
rsync: [generator] chown "/var/www/html/3rdparty/doctrine/cache/lib" failed: Operation not permitted (1)
rsync: [generator] chown "/var/www/html/3rdparty/doctrine/cache/lib/Doctrine" failed: Operation not permitted (1)
rsync: [generator] chown "/var/www/html/3rdparty/doctrine/cache/lib/Doctrine/Common" failed: Operation not permitted (1)
rsync: [receiver] chown "/var/www/html/3rdparty/aws/aws-sdk-php/src/data/cloudfront/2020-05-31/.api-2.json.php.VVyVv1" failed: Operation not permitted (1)
rsync: [receiver] chown "/var/www/html/3rdparty/aws/aws-sdk-php/src/data/cloudfront/2020-05-31/.paginators-1.json.php.DiLrlZ" failed: Operation not permitted (1)
rsync: [receiver] chown "/var/www/html/3rdparty/aws/aws-sdk-php/src/data/cloudfront/2020-05-31/.smoke.json.php.pwuGv2" failed: Operation not permitted (1)
rsync: [receiver] chown "/var/www/html/3rdparty/aws/aws-sdk-php/src/data/cloudfront/2020-05-31/.waiters-1.json.php.oWgzLZ" failed: Operation not permitted (1)
rsync: [receiver] chown "/var/www/html/3rdparty/aws/aws-sdk-php/src/data/cloudfront/2020-05-31/.waiters-2.json.php.oyHzh1" failed: Operation not permitted (1)
rsync: [receiver] chown "/var/www/html/3rdparty/aws/aws-sdk-php/src/data/cloudhsm/2014-05-30/.api-2.json.php.q9gc20" failed: Operation not permitted (1)
rsync: [receiver] chown "/var/www/html/3rdparty/aws/aws-sdk-php/src/data/cloudhsm/2014-05-30/.paginators-1.json.php.NCqlOY" failed: Operation not permitted (1)
rsync: [receiver] chown "/var/www/html/3rdparty/aws/aws-sdk-php/src/data/cloudhsmv2/2017-04-28/.api-2.json.php.mlX0Y1" failed: Operation not permitted (1)
rsync: [receiver] chown "/var/www/html/3rdparty/aws/aws-sdk-php/src/data/cloudhsmv2/2017-04-28/.paginators-1.json.php.HFxAd2" failed: Operation not permitted (1)
rsync: [receiver] chown "/var/www/html/3rdparty/aws/aws-sdk-php/src/data/cloudhsmv2/2017-04-28/.smoke.json.php.7nseHZ" failed: Operation not permitted (1)
rsync: [generator] chown "/var/www/html/3rdparty/doctrine/cache/lib/Doctrine/Common/Cache" failed: Operation not permitted (1)
rsync: [generator] chown "/var/www/html/3rdparty/doctrine/cache/lib/Doctrine/Common/Cache/Psr6" failed: Operation not permitted (1)

What was expected:

Nextcloud should be up and running.
But instead it fails it's liveness probe etc.

###Full configuration and tests
AWS: "Prevent root access by default*" in EFS is NOT enabled. Providing the same claim to a postgres-cluster works fine.

values.yaml:

image:
  repository: nextcloud
  pullPolicy: IfNotPresent

ingress:
  enabled: true
  className: nginx
  annotations:
    nginx.ingress.kubernetes.io/from-to-www-redirect: "true"
  nginx.ingress.kubernetes.io/proxy-body-size: 4G
  kubernetes.io/tls-acme: "true"
  cert-manager.io/cluster-issuer: letsencrypt-production
  tls:
    - secretName: tls-letsencrypt-production
      hosts:
        - www.nextcloud.example.io
  labels: {}
  path: /
  pathType: Prefix

nextcloud:
  host: www.nextcloud.example.io
  password: changeme
  username: someUser

internalDatabase:
  enabled: true
  name: nextcloud

persistence:
  enabled: true
  annotations: {}
  existingClaim: nextcloud-claim
  nextcloudData:
    enabled: true
    existingClaim: nextcloud-data-claim

claims.yaml:

apiVersion: v1
kind: PersistentVolumeClaim
metadata:
  name: nextcloud-claim
spec:
  accessModes:
    - ReadWriteMany
  storageClassName: efs-sc
  resources:
    requests:
      storage: 8Gi
---
apiVersion: v1
kind: PersistentVolumeClaim
metadata:
  name: nextcloud-data-claim
spec:
  accessModes:
    - ReadWriteMany
  storageClassName: efs-sc
  resources:
    requests:
      storage: 8Gi

Start by reproducing the Helm installation on AWS EKS with the provided values.yaml, PVCs, and EFS-backed storage, then inspect the chart's persistence-related configuration and pod logs. Done means the Nextcloud pod starts successfully, its liveness probe passes, and the reported rsync chown failures no longer prevent startup.