This question came to my mind this afternoon while trying to figure out a possible way to tackle de Polignac's conjecture, which states that every even positive integer can be written as the difference of two consecutive primes in infinitely many ways, and is of course related to About a possible generalization of Green-Tao's theorem.

So here it comes: Can **the proof** of Green-Tao's theorem be quite naturally slightly modified to show **unconditionally** that the sequence of $1$-central numbers, i.e positive integers equal to half the sum of two consecutive odd primes, contains arbitrarily long arithmetic progressions?

Many thanks in advance.

notprime for $0<i<k$ (an upper bound on the characteristic function of primes) and the $n\pm k$isprime (a lower bound). $\endgroup$