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The answer is no, if you believe the Hardy-Littlewood $k$-Tuple Conjecture. Let $\pi_{k}(x)$ denote the number of primes $p\leq x$ such that $p+2k$ is also prime. Then the conjecture predicts $$ \pi_{k}(x) \sim C(k) \int_{2}^{x} \frac{dt}{(\ln t)^{2}}, ​ $$ where $$ C(k) = 2\prod_{p>2} \frac{p(p-2)}{(p-1)^2} \prod_{\substack{p\mid k\\ p>2}} \frac{p-1}{p-2}. $$