Docker 26 added the environment variable DOCKERD_ROOTLESS_ROOTLESSKIT_DISABLE_HOST_LOOPBACK=false which allows a rootless container to connect to a service running on host: https://docs.docker.com/engine/release-notes/26.0/#new

I still do not find any way to ping the host from the rootless container...

BUT the rootless container connects successfully to the websocket service running on the host identified by ip address 10.0.2.2.