A condition for being a prime: $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$

If $\;p=m+n$ where $p\in\mathbb P$, then $m,n$ are coprime, of course. But what about the converse?

Conjecture:

$p$ is prime if $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$

Tested (and verified) for all $p<100000$.

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