This is an answer to the last part of you question: has this conjecture been proposed already?. My answer is: no, not to the best of my extremely limited knowledge.
However, as mentioned in the comments, this statement is a weakening/specialisation of Opperman's conjecture. So a refutation/counter-example of this conjecture would lead to a refutation of Opperman's conjecture, which would be a massive result in mathematics, so I wouldn't expect that to come too easy.
Now the question is, how hard is this special case relative to the overall conjecture? More specifically, how much easier does the conjecture get when we restrict $n$ to only primes? Are there any tricks we can exploit, knowing that $n$ is prime, that we couldn't exploit in the general case? The disappointing answer is: I don't know. It could be that there is a relatively elementary proof for this conjecture, exploiting the primeness of $n$, while Opperman's conjecture requires much more "heavy machinery" to prove. Or it could well be that Opperman's conjecture is false and this one true. Or it could be that both are false, but the counter-examples are way beyond our present computational power. Or maybe we'll find a counter-examle to Opperman's conjecture, but this one will remain unsolved for another century. There are many such possibilities.
In any case, I would advise deeper research into Opperman's conjecture for more answers to this question. Perhaps an expert in that area will know the answer to this question.