The refined Malle conjecture for the Heisenberg group over $\mathbb{F}_4$

Let ${\rm Heis}_4$ be the order-$64$ Heisenberg group over $\mathbb{F}_4$, viewed in its regular permutation representation ${\rm Heis}_4\leq S_{64}$. Let $C_0$, $C_\beta$, $\alpha_0$, and $\alpha_\beta$ be the constants defined by

$$C_0=\alpha_0\prod_{p>2}\left(1+\frac{9}{p}+\frac{9}{p^{3/2}}\right)\left(1-\frac{1}{p}\right)^9,$$ $$C_\beta=\alpha_\beta\prod_{p>2}\left(1+\frac{9}{p}+\frac{3}{p^{3/2}}\right)\left(1-\frac{1}{p}\right)^9,$$ $$\alpha_0=\frac{313+54\sqrt{2}+54\sqrt[4]{2}+9\sqrt{2}\sqrt[4]{2}}{2^{71}\cdot3^4\cdot5},\qquad \alpha_\beta=\frac{247+30\sqrt[4]{2}+36\sqrt{2}-3\sqrt{2}\sqrt[4]{2}}{2^{71}\cdot3^3\cdot5}.$$

Refined Malle conjecture for ${\rm Heis}_4$. The group ${\rm Heis}_4$ fails to satisfy independence of local events, and

$$\#\left\{K/\mathbb{Q}\ \text{Galois}:\begin{matrix}|\operatorname{disc}(K/\mathbb{Q})|\leq X,\\operatorname{Gal}(K/\mathbb{Q})\simeq{\rm Heis}_4\end{matrix}\right\}\sim\frac{1}{2}(C_0+C_\beta)X^{1/32}(\log X)^8.$$

This gives an explicit leading-constant prediction for Galois ${\rm Heis}_4$-extensions of $\mathbb{Q}$ ordered by discriminant. It is conditional on the conjectural framework concerning independence of local events and exhibits a sum of two Euler products rather than a single Euler product.