The answer is no, if you believe the [Hardy-Littlewood $k$-Tuple Conjecture][1]. 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
$$
C(k) = 2\prod_{p>2} \frac{q(q-2)}{(q-1)^2} \prod_{\substack{p\mid k\\ p>2}} \frac{q-1}{q-2}.
$$


[1]: https://mathworld.wolfram.com/k-TupleConjecture.htm