You can't stop them coming in but you can isolate them and then flush the conversations. This would be done by creating a work queue in a higher position to where your facebook traffic lands with the FB page and private tags in the ALL section. See example image below. This work queue should not be granted to teams or users unless you actually want them to see this traffic. This step effectively hides traffic from users. Then create a priority rule that is exactly the same, again positioned above where you set priorities for other facebook traffic, and set a priority that has a flush time on it (you can submit a ticket to support if you dont have one). This will ensure the conversations are closed, if you have a spare priority I would recommend using that and having it set to the absolute minimum time so that posts can be flushed quickly. Now, the limitations and the reason the flush should be as short as possible, of this approach is attachment. Where a user has both private and publicly messaged your page the conversations will be attached to each other. This means that public posts may be flushed. It is also possible for agents to see private messages in queues they don't have access to view via the Author profile, where it lists previous/current conversations in the platform, and via the related posts (where two open conversations exist).
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