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Is Owner profile always making network connections, just as many as if it were in the foreground?
Yes.
If the Owner profile has a VPN connected, do connections made in the owner profile route through the owner profile VPN while you are in a secondary profile?
Yes.
Simlarly, if the owner profile has no VPN connected, and you're in a secondary profile (lets say it has a VPN connected), the owner profile will still be making connections to your ISP?
Yes.
Do apps in the Owner profile drain battery while you are in a secondary profile?
Yes.
Would it make sense to disable any battery-draining apps?
You can set their battery mode to "restricted".
What other connections or activity is made from the owner profile while you are in a secondary profile?
The same stuff as if you were idling on the home screen with the profile open.