Facebook Comment Plugin Synced with Page?

I'm using the Facebook Comment plugin for comments on my site. I was hoping that comments made to links shared on my Facebook Page would also show 开发者_运维技巧up on my website.

For example, if I share a link to a blog post on my Facebook Page, and someone comments directly to that shared link, that comment does not also appear on the website.

Am I missing something? Is there a way to sync these two?

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I think comments are self-contained. You could maybe use the API to build something that shares comments on your page with comments on your blog.