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It’s a new conjecture, all right. But it’s clearly false.
Consider n=4. Then p=5, q=3, k=1. But 5+1 and 3+1 are not primes.
All you number theorists out there, I think your jobs are safe for the time being.
It’s not saying its proof is true though, it just posed a conjecture. Proving it would be a whole different ballgame.