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 $(a, q) = 1$ and $q$ is large as a function of $X$?
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 $$|\{(x, y, z): x, y, z\in \mathbb{P}_{a, q}\cap [1, X]\text{ and }x+z=2y\}|\gg \frac{1}{\phi(q)^3}\frac{X^2}{(\log X)^3}.$$