Equidistribution of prime pairs among coset lattice congruence classes

Let $z$ and $q$ be fixed, let $\mathcal{N}_q(a,b)$ denote a coset lattice congruence class for prime pairs of gap $z$, and let $\pi_z(x;\mathcal{N}_q(a,b),q)$ count the pairs $(p_i,p_{i+z})$ with $p_i,p_{i+z}\leq x$ in that class. Write $\phi(q)=\#\{\mathcal{N}_q(a_i,b_i)\}$. Equidistribution conjecture. For some constant $\mathcal{D}(z)>0$,

$$\pi_z(x;\mathcal{N}_q(a,b),q)\sim \frac{x\mathcal{D}(z)}{\phi(q)\log^2x}.$$

This asserts equidistribution of prime pairs with a fixed gap among the relevant coset lattice congruence classes, refining the known order of growth $\pi_z(x)\asymp_z x/\log^2x$. Establishing this asymptotic is stated as the theme of the paper; the supplied text gives no evidence that it has been proved.