Naive fair-counting adaptation of Malle's conjecture

Let $G$ be a non-trivial finite group and let $k$ be a number field. Write $\mathfrak{f}(K/k)$ for the product of the primes of $k$ that ramify in $K$. For $g\in G-\{\operatorname{id}\}$, let $\operatorname{Cl}(g)$ be its conjugacy class, and write $g\sim h$ when $\operatorname{Cl}(g)$ and $\operatorname{Cl}(h)$ are equivalent under the cyclotomic action of $k$. The fair-counting adaptation of Malle's conjecture. There exist an integer $b(G,k)\geq 0$ and a real number $c(G,k)>0$ such that

$$\left|\left\{K/k:\operatorname{Gal}(K/k)\cong G,\ N_{k/\mathbb{Q}}(\mathfrak{f}(K/k))\leq X\right\}\right|\sim c(G,k)X(\log X)^{b(G,k)},$$

and $b(G,k)=-1+n$, where $n$ is the number of equivalence classes of $G-\{\operatorname{id}\}$ under $\sim$. This is a naive modification using the product of ramified primes as a fair counting function; the paper's abstract states that it is not true in general.