Publish from Elgg to Facebook

I have been looking around for any plugin connecting Elgg and Facebook accounts in the way we can post Twitter updates to a Facebook page. Given a public group in Elgg where every post is open to any non-registered user, would it be possible to automatically publish it in a Facebook page. I imagine that this is very tricky because in a group there can be many users with writing permissions but, has anybody tried to do it from users' profiles. Many thanks for any hint.