A solution could be to give people a window of time (say, 5 years) in which to move their coins from old addresses to new, quantum resistant addresses. Then miners would stop including transactions from non quantum addresses in their mined blocks. (a soft fork can do that). It is censorship in some way, but with a long time frame for people to adapt to it.
during this time, wallets could start warning users that they are using outdated addresses, and they should migrate ASAP.
Any miner running an older version of Bitcoin that mines those coins being spent would lead to a chain split (one chain they are spent, the other they are locked) - aka hard fork. Such a change is not backwards compatible.
Chain split != hard fork
Clients would follow the chain with the most PoW
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