Revised Malle conjecture for number fields

Let $N\subseteq S_n$ be a finite transitive subgroup and let $k$ be a number field. For each normal subgroup $G\leq N$ with abelian quotient $N/G$ and $a(G)=a(N)$, let $b(G,N,k)$ be the maximum of $b_{\varphi}(G,N,k)$ over surjective homomorphisms $\varphi\in\operatorname{Hom}(G(k^c/k),N/G)$, where $k^c$ is the maximal cyclotomic extension of $k$ and $b_{\varphi}(G,N,k)$ counts the orbits of minimal-index conjugacy classes of $G$ under the corresponding $\varphi$-twisted action. Define

$$\nb(N,k)=\max\{b(G,N,k):N/G\text{ is abelian and }a(G)=a(N)\}.$$

Revised Malle conjecture. Fixing $N$ and $k$, one has

$$\mathcal{Z}_N(k,X)\asymp X^{a(N)}(\log X)^{b(N,k)-1}.$$

The revision is proposed to correct the logarithmic exponent in Malle's conjecture for number fields, in light of counterexamples to the original formulation. The source proposes this statement based on the paper's results, but gives no resolution of the revised conjecture.