Only end-users can start a conversation with the business. Once the conversation is finished, it's not possible for the business to send new messages. If an end-user does a story mention of your business, then this will allow the business to start direct message communication with the end-user.
If the business receives story replies from their followers, then the business can start a conversation using the reply. If a business receives a direct message from its followers, then it can start a conversation using it. If an end-user were to post a comment (public) then the business can reply to this follower privately.