Erdős problem 1209

Let $A=\{a_1<a_2<\cdots\}$ be a sequence of integers which tends to infinity sufficiently fast. If there is an $n$ such that all $n+a_k$ are primes then must there exist infinitely many such $n$?

Erdős [Er80] wrote 'unless I overlook a trivial way of getting a counterexample these questions are quite hopeless'. There is indeed a trivial counterexample (a variant of the construction in [erdosproblems.com/429]): define $a_1=2$ and for $k\geq 2$ let $a_k>a_{k-1}$ be a prime such that $a_k+k\equiv 0\pmod{q_k}$, where $q_k$ is some prime not dividing $k$. This sequence can be made to grow arbitrarily fast

See also [erdosproblems.com/429] and [erdosproblems.com/1102].