A while ago, I spent some time to make LXC run in a docker container. The idea is to have a statefull system managed by LXC run in a docker environment so that management (e.g. Volumes, Ingress and Load Balancer) from K8S can be used for the LXC containers. I still run a few desktops which are accessible by x2go with it on my kubernetes instances.

https://github.com/micw/docker-lxc