The full moment conjecture for quadratic Dedekind zeta functions

Let $\mathbb{K}$ be a quadratic extension, let $\chi$ be the associated real Dirichlet character, and let $a(k)$ be the arithmetic factor defined by the Euler product in the paper. The quadratic Dedekind zeta moment conjecture. One has

$$I_k(T)\sim a(k)L(1,\chi)^{2k^2}\left(\frac{G(k+1)^2}{G(2k+1)}\right)^2\left(\log T\cdot\log qT\right)^{k^2}.$$

This is the predicted full asymptotic for moments of quadratic Dedekind zeta functions, combining the arithmetic Euler-product factor with the two zeta-type random-matrix factors. The paper proves the corresponding splitting statement in the second-moment case $k=1$, but the full formula is conjectural.