Are there arbitrarily long arithmetic progressions in the sequence of twin primes?

The main result of the paper is that assuming that the level $\theta$ of distribution of primes exceeds 1/2, then there exists a positive $d\leq C(\theta)$ such that there are arbitrarily long arithmetic progressions with the property that $p'=p+d$ is the next prime for each element of the progression. If $\theta>0.971$, then the above holds for some $d\leq 16$.