Currently 'Boost for Lemmy' doesn't show any real-time notification. The minimum check interval can be set to 30 minutes and that's also inconsistent. There should be an option or by default it should show real-time notification. This issue was also present in 'Boost for Reddit'.
Lemmy would need to push notifications to you from their servers for that. Periodically checking your inbox for messages is the best any app can do on their end. It was the same with Reddit, where they would only push notifications to the official app, and anything else would be limited to checking on its own.
Increasing the frequency of checks would also have an impact on your device's battery life and increase traffic for your instance.
Google and Apple both provide push notification services to avoid the polling problem.
But that would require each and every instance to register with each and every app for it to work, which is definitely not going to happen.