Let $d_n=p_{n+1}-p_n$, where $p_n$ is the $n$th prime. Let $r(x)$ be the smallest even integer $t$ such that $d_n=t$ has no solutions for $n\leq x$.

Is it true that $r(x)\to \infty$? Or even $r(x)/\log x \to \infty$?