The statement in the original post is true. This follows from a recent result of Pintz, which is a common generalization of the Green-Tao theorem and the Maynard-Tao theorem:

Theorem. Let $m\geq 2$ be an integer. Then for any sufficiently large admissible set $\mathcal{H}$, there exist $h_1,\dots,h_m\in\mathcal{H}$ such that the set $$\{n\in\mathbb{N}:\ \text{$n+h_1,\dots,n+h_m$ are consecutive primes}\}$$ contains arbitrary long arithmetic progressions.

Indeed, the special case $m=2$ is already sufficient to derive the statement in the original post.

The statement in the original post is true. This follows from a recent result of Pintz, which is a common generalization of the Green-Tao theorem and the Maynard-Tao theorem:

Theorem. Let $m\geq 2$ be an integer. Then for any sufficiently large admissible set $\mathcal{H}$, there exist $h_1,\dots,h_m\in\mathcal{H}$ such that the set $$\{n\in\mathbb{N}:\ \text{$n+h_1,\dots,n+h_m$ are consecutive primes}\}$$ contains arbitrary long arithmetic progressions.

Indeed, the special case $m=2$ is already sufficient to derive the statement in the original post.

P.S. Note that the main theorems in Pintz's paper are stated with a typo: the requirement that $p_i^*$ is prime should be dropped (it is the $n$ above). I notified János Pintz, and he agreed with this correction.