Garmin connect must run in the background for va3m to be connected?

Are you connected both with the app and also the phone's bluetooth? You should be.

On my Vivoactive, not only does it stay connected but it automatically syncs when I complete an activity even if the app is closed.

It looks as if the bluetooth ia connected with the phone since I see it on my bluetooth list

Then I see no reason why you should lose connection. Does your watch face show the bluetooth icon when connected?

Yes only when garmin connect is running. When I close it after few seconda the bluetooh icon disappear.

I wonder if an energy-saving setting on your phone is interfering.

I started to think it's something with the Honor 9 (Android 8)bluetooth's setting which I haven't found yet.

After restoring to factory setting the watch started to keep the bluetooth connection even without starting garmin connect. But when I open Garmin Connect and close it the connection drops. I need to restart my phone in order to restore the bluetooth connection without opening the Garmin Connect.
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I think I found the solution in my honor 9, under settings->apps->settings (flatong circle at the bottom of the screen) ->special access->notification access
-> turn on the Android system and also the Smart notification (it has the garmin connect icon).
When those two turned on I don't need to kepp the Garmin connect apps running.