the difficult part is dealing with the 5+ million vulnerable coins (p2pk outputs, outputs sitting in reused addresses, shared xpubs, etc). implementing a post-quantum signature scheme alone doesn't address the fact that 1/3 of the supply is vulnerable to theft. people need to voluntarily move their coins to quantum-safe addresses for the fork to be effective. that could take a few years, based on the adoption rate of segwit.
Indeed. The question of what to do with the coins that are
not moved to quantum-proof addresses is a huge problem.
From my amateurish perspective, it seems to me that if the problem couldn't be solved in time, and it came to a choice between either
(a) burning anything that hasn't been moved, or
(b) leaving them there to be scooped up by a QC
... then I think option (a) is far preferable.
You can't just soft-fork to a situation where some bitcoins are quantum resistant and some aren't; (b) could lead to another gox or worse.
A hard-fork option (a) would still be hugely contentious but if it comes down to a question of bitcoin's survival, it's the better option. Either way you're never going to get a consensus, and there would likely be a serious* chain-split.
*serious, not like BCH.