Email Notifications set to "As Soon As Possible" - Not receiving

I'm set to Do Not Disturb with a handful of people tagged as "Priority Access", and I have my "Missed Activity Emails" set to "as soon as possible." The point is so that I can get a chat message from select folks, while any chat message from other people will go straight to email so I can review it and see if I need to respond immediately or not. 

Someone not on my Priority list sent me a chat; 10 minutes later I see the chat but no email. 

What am I missing? 

Thank you!