Pair-prime distribution conjecture for imaginary quadratic norm forms

Let $f(a,b)=a^2+Db^2$ be the norm form in the imaginary quadratic field $\mathbb{Q}(\sqrt{-D})$. For positive $a,b,c,d$, let $p=f(a,b)$ and $q=f(a+c,b+d)$ be primes less than a large number $M$, with slopes $b/a<T$ and $d/c<T$. Put $\theta=\tan^{-1}(T\sqrt{D})$.

Imaginary quadratic pair-prime conjecture. The distribution is

$$D(\theta)=\int_0^\theta\int_0^\theta\frac{(\alpha-\beta)-\sin(\alpha-\beta)\cos(\alpha-\beta)}{2\sin^3(\alpha-\beta)}\,d\alpha\,d\beta.$$

This is the pair-prime analogue of the one-prime slope distribution and is related, up to a constant, to the continuous multiple Dedekind zeta value at $(2,2)$. Numerical tests are reported for Gaussian and other imaginary quadratic fields.