Opening the host's browser automatically after Dev Container startup

Note: I had previously posted this question on StackOverflow, but not received any response.

I'm using the Dev Containers extension v0.362.0 to do my development work. I would like to automatically open a specific website in the host's web browser after the container has started up.

After the container has booted, I can manually open a terminal and run:

xdg-open http://...

or

code --openExternal http://...

As expected, the host's web browser will open the requested page. The xdg-open command works because the remote extension seems to be setting the BROWSER environment variable, which then forwards the request to VS Code itself.

So far, so good!

But I cannot achieve the same when using a postStartCommand or postAttachCommand in devcontainer.json, i.e.,

  "postStartCommand": {
    "open-website": "code --openExternal http://..."
  },

and

  "postAttachCommand": {
    "open-website": "code --openExternal http://..."
  },

do not work: nothing seems to happen.

I figured that when these commands are run, the BROWSER variable is not (yet?) set.

I'm wondering if I'm perhaps missing something essential here? Any suggestions would be greatly appreciated.