Von Mangoldt reformulation of prime-pair equidistribution

Let $z$ and $q$ be fixed, let $\mathcal{N}_q(a,b)$ denote a coset lattice congruence class for prime pairs $(p_i,p_{i+z})$, and let $\Psi_z(x;\mathcal{N}_q(a,b),q)$ be the corresponding weighted counting function. Write $\phi(q)=\#\{(a,b):(p_i,p_{i+z})\in\mathcal{N}_q(a,b)\}$. Weighted equidistribution conjecture. For some constant $\mathcal{D}(z)>0$,

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

This is presented as a reformulation of the preceding prime-pair equidistribution problem using the von Mangoldt function. The supplied text does not establish the asymptotic, so its resolution is not indicated.