Here's how the Facebook page live video solution works.

Your live video must have a title or description.
If your page is live, the live video will be displayed.
SociableKIT can detect a live video in 1-5 minutes after you go live on Facebook.
If your page is not live, it will display this message: We are not live right now.
If your page is not live, and you checked the "Show latest video if not live" option, it will show your latest video.

Please note that it won't auto-play on phones. But there will be a Live status and a play button like this:

Once the user tapped the play button, it will play in full screen, with a Live broadcast label, like this:

That way, it is better for the user experience.

You need to follow the following rules as well.

You must be an admin of your target Facebook page.
Your Facebook page and it videos must be set to public.
There must be no age and country restrictions on your Facebook page.
Your Facebook page must have at least 25 likes. This rule applies to some Facebook pages only.
Make sure your video can be embedded. You can edit your video on Facebook to do this.

Go to Distribution > Check Allow Embedding > Click Save button.
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