Hello,

Let's say that an integer $n$ is $k$-primal if $k$ is its smallest primality radius (i.e non negative integer $r$ such that both $n-r$ and $n+r$ are primes).
I think that for every positive integer $m$ and every non negative integer $k$, there exists an arithmetic progression made of $m$ $k$-primal integers.

Has such a generalization of Green-Tao's theorem been considered so far?

If not, is there a heuristics that would make it quite likely?

Thanks in advance.