I just connected, took a few shots, then turned off the iPhone screen and took a few more shots. The location was saved in the photos with the screen off, so it seems to work ok. While I was looking at the images on my iPhone, I saw the camera screen pop up a message to say it had disconnected from the app. I woke up the iPhone and while wifi was now off, BT was still on. So maybe it will work ok to record GPS data with the screen off.
The app is not as smooth running as I'd like, but certainly not as bad as I thought based on complaints I've seen. The new LUMIX Lab app looks to be faster so hopefully there are better times ahead.
Here is the test I did.
Gear and software
Camera : S1R running firmware 2.0
Phone : iPhone 11 Pro running iOS 17.5.1
LUMIX Sync app : version 2.0.12
Setup and basic test
1) Connect camera to LUMIX App with BT only
2) Set camera to geotag
3) Take photo - location is recorded in file
Camera sleep test - all good
1) Keep app alive on iPhone
2) Allow camera to sleep
3) Wake camera up - it reconnects to BT after a short time
4) Take photo - location is recorded in file
App in background test - all good
1) Put app in background, but interact with phone to keep it awake
2) Don't let camera sleep
3) Camera remains connected
4) Take photo - location is recorded in file
iPhone sleep test - all good
1) Put app in background, allow iPhone to sleep (turn off screen)
2) Don't let camera sleep
3) Camera remains connected
4) Take photo - location is recorded in file
iPhone sleep and camera sleep - fail
1) Put app in background, allow iPhone to sleep (turn off screen)
2) Allow camera to sleep
3) Wake camera up - it DOES NOT reconnect
4) Take photo - location is not recorded in file
Unfortunately, the last case is actually the most common use case!