YMMV, I'm genuinely interested how others see it. Please, try to explain if your opinion differs. I might be wrong with my own opinion and if so, I'd like to learn why!
I think it is technically possible to prove, that you are the real solver. Which means, that everyone would know, if coins were stolen or not. Solvers just don't use such proofs, but it can be done.
First, any solver can deposit any coins into its own, strong key, and a proof of being the one, who solved the puzzle. The output Script is:
<strongPubkey> OP_CHECKSIGVERIFY OP_RIPEMD160 <puzzleHash> OP_EQUAL
Then, the solver can wrap it behind P2WSH, and wait for getting enough confirmations. After a while, a solver can move the coins from this Script, to the desired destination of the puzzle. It can even be done in the same transaction, which would sweep the puzzle, if desired.
And then, it won't prevent coins from being stolen, but it would prove to the outside world, if they were stolen or not.
Another thing is proving knowledge of some public key with ZK-proofs, DLEQ proofs, or something similar. It can be done, and it could be battle-tested on puzzles, before introducing any "quantum-resistant" things globally.