Explicit leading constant for odd nilpotent groups

Let $G$ be an odd-order nilpotent group, with $b_{(H,\phi)}(G)$ and the local factors $\alpha_1,\alpha_2,\alpha_3$ as defined in the surrounding discussion and in Proposition. Explicit constant conjecture. The asymptotic conjecture holds with

$$c_{(H,\phi)}(G)=\frac{1}{\Gamma(b_{(H,\phi)}(G))}\times\alpha_1\times\alpha_2\times\alpha_3.$$

This gives a predicted Euler-product-style expression for the leading constant in the odd nilpotent case, but the supplied text does not state a separate resolution of this prediction.