I also tried running with the H10 polar strap, and mid-run switched the connection from watch to polar app. Polar app reads it correctly. Garmin won't. I don't have any garmin branded HRM to test, but my understanding was that as long as I use ANT for transmission the manufacture of the strap should not make a difference.
My Garmin Device Will Not Pair or Display Data From My Chest Heart Rate Monitor | Garmin Customer Support. Eqw8.