Continuous multiple Dedekind zeta conjecture for prime slopes

Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $f(a,b)=N(a\mu_1+b\mu_2)$ be the associated irreducible homogeneous norm form. For positive $a,b$ with $0<b/a<T$, consider primes $p=f(a,b)<M$. The continuous multiple Dedekind zeta value is denoted by $\zeta^{\operatorname{continuous}}_{K,C_T}(2)$.

Continuous multiple Dedekind zeta conjecture. For fixed large $M$, as $T$ varies, the distribution of the number of represented primes is given by

$$\zeta^{\operatorname{continuous}}_{K,C_T}(2).$$

The associated change-of-variables computation relates this quantity to the integral slope distribution $\int_0^T[\sqrt[n]{f(1,t)}]^{-2}\,dt$, connecting prime representation statistics with multiple Dedekind zeta values.