Average Dirichlet-series conjecture for quadratic prime pairs

For $r\in\mathbb{N}$, let $\theta^*_{2r}(x)$ count primes $p\leq x$ such that $p^2\pm2r$ is prime, and define, for $s=\sigma+i\tau$,

$$D^*_{2r}(s)=\sum_{p,\,p^2\pm2r\,\operatorname{prime}}\frac{\log^2 p}{p^{4s}}=\int_1^\infty x^{-4s}\,d\theta^*_{2r}(x).$$

Average quadratic prime-pair Dirichlet-series conjecture. For $\sigma>1/4$ and $N\to\infty$,

$$\frac{1}{N}\sum_{r=1}^N D^*_{2r}(s)=\frac{2+o(N^{-1/2})}{4s-1}+H^N_2(s),$$

where $H^N_2(s)$ is analytic and has good boundary behavior as $\sigma\searrow1/4$. This conjecture is introduced as a Dirichlet-series analogue supporting the averaged Bateman–Horn prediction; its asserted boundary behavior and asymptotic remain open in the supplied source.