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All users prefer the longest chain. In the case of two solutions being broadcast at about the same time, then each miner will probably prefer the solution they received first, and will base their solution attempts off of that chain. It is very unlikely that both chains will each have a new block broadcast at the same time. Whichever chain gets a new block first will be the winner and everyone will switch to that chain, abandoning the now-orphan chain.

The two chains will fall out of sync before they're finished. Chance of all 12 proofs being completed at exactly the same time and broadcast to groups of identical computing power every step of the way is very small.

Murphy's law: "If it can happen, it will happen"

Some probabilities approach zero faster than the universe can approach heat death.

What if it involves hash collisions?

No SHA2 hash collisions have ever been found.

What's the chance of that happening 12 times in a row?

(Rhetorical question, you'll find the answer in the last section of the bitcoin whitepaper.)