I was curious about this point when discussion came up on HN just recently. I don't see how you could "assign" each non-QR address a new quantum resistant address unless they "claim" it themselves somehow. What can possibly happen to an uneducated mom-and-pop bitcoin holder who never takes up their claim? Someone else who cracks their private key would be in an identical position to them w.r.t authenticating themselves and doing such a claim first - thus it becomes a race