Let $\mathbb{P}_{a, q}$ denote the set of primes congruent to $a$ modulo $q$. Are there any estimates for the number of $3$-Arithmetic Progressions in the set $\mathbb{P}_{a, q}\cap [1, X]$, where $(a, q) = 1$? 

For $q$ at most a constant, such an estimate can be given thanks to Green [1]. One can take a slightly better $q$ thanks to the work of Helfgott and Roton [2]. Both works deal with a much more general setting of an arbitrary dense subset of primes. Can one gain in how large of a modulus $q$ is admissible if one restricts specifically to $\mathbb{P}_{a, q}\cap [1, X]$?

To be more precise, I want to know how large of a modulus $q$ one can take as a function of $X$ such that one has a result of the kind
$$|\{(a, b, c): a, b, c\in \mathbb{P}_{a, q}\cap [1, X]\text{ and }a+c=2b\}|\gg \frac{1}{\phi(q)^3}\frac{x^2}{(\log x)^3}.$$

[1] https://arxiv.org/pdf/math/0302311

[2] https://arxiv.org/abs/0912.1842