The Artin-Schreier-Witt counting conjecture for noncyclic abelian p-groups

Let $F$ be a global function field of characteristic $p$, and let $G$ be a finite noncyclic abelian $p$-group. Write $Z(F,G;X)$ for the number of Artin-Schreier-Witt extensions $E/F$ with Galois group $G$, ordered by discriminant norm, and let $a_p(G)$ and $\beta(F,G)$ be the quantities governing the predicted power and logarithmic exponents. Artin-Schreier-Witt counting conjecture. I conjecture that there is a constant $c(F,G)$ such that

$$Z(F,G;X) \sim c(F,G) X^{a_p(G)} \log(X)^{\beta(F,G)-1}.$$

This predicts that the defect appearing in the lower-bound analysis does not affect the power of $X$ in the asymptotic. The surrounding discussion gives an upper bound with exponent $d_p(G)$ and explains that the conjectured exponent $a_p(G)$ is expected to be sharper; the conjecture remains unresolved in the stated generality.