Refined moment and asymptotic conjecture for admissible 2-group tuples

Let $(G_i,H_i,T_i)$, for $1\leq i\leq k$, be admissible tuples of finite $2$-groups and generating sets. Let $f_{T_i}(K)$ be the associated refined counting functions, let $c_{T_i}$ be the corresponding constants, and let $\boldsymbol{E}_{1}^{\pm}$ denote the average over quadratic fields of positive or negative discriminant. Refined moment and asymptotic conjecture. For every positive integer $k$,

$$0<\boldsymbol{E}_{1}^{\pm}\left(\frac{\prod_{i=1}^{k}f_{T_i}}{\prod_{i=1}^{k}c_{T_i}^{\omega(D_K)}}\right)<\infty.$$

If all tuples equal $(G,H,T)$, these moments determine a distribution on the values of $f_T$. Furthermore, there is a constant $C(G,H,T)$ such that

$$\sum_{K,0<\pm D_K<X}\prod_{i=1}^{k}f_{T_i}(K)\sim C(G,H,T)X(\log X)^{\prod_{i=1}^{k}c_{T_i}^{k}-1}.$$

This refines the preceding conjecture by incorporating generating sets and the corresponding refined counting functions; the source presents it as a heuristic extension of asymptotic predictions for even groups.