Arithmetic progressions of twin primes of every length

An arithmetic progression of twin primes with $l$ terms is a sequence $a+kb$, for $k=0,1,\ldots,l-1$, whose terms are first members of twin-prime pairs, where $a$ and $b$ are positive integers.

Twin-prime progression conjecture. For every natural number $l$ with $l>2$, there exist arithmetic progressions of twin primes with $l$ terms.

This is an analogue of the Green–Tao phenomenon for configurations formed by twin primes. It remains open because no theorem currently establishes such progressions for every length.