I understand that this is kind of a broad question, but if no affirmative proof is known, can anyone give a counterexample?
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If you're talking about linear congruences of the form $x \equiv a \bmod b$, then the answer is yes.
The key point is that $\gcd(a,b)$ divides every solution $x$. Thus:
Therefore, the only case when there is more than one prime solution is the first case, in which there are an infinite number of prime solutions.