Check. Thanks for messaging me btw 🙂. Not using battery saver. Also got this reply/mention alert through Tusky but not Fedilab and phone was 100%. I'm looking closely but I'm not sure if my eye is missing something.
Yes, the keep background process when the app is closed is checked under the notification service also. I downloaded Fedilab through FDroid (Tusky through Google Play) but I don't see why that would affect the notifications.