Rarity of fixed inertia-type prime counts in number fields

Let $1\leq G\leq S_n$ be a transitive permutation group, let $k$ be a number field, and let $\mathcal{S}$ be the set of number fields $(K/k,\psi)$ whose Galois closures $(\hat{K}/k,\psi)$ are $G$-extensions and for which $K=\hat{K}^{G_1}$, where $G_1$ is the stabilizer of $1$. For a nonempty subset $\Omega\subseteq G$ closed under invertible powering, let $\mathbf{1}_{(\Omega,r)}(K)$ indicate that exactly $r$ primes $p\nmid |G|$ have inertia subgroup $I(p)$ intersecting $\Omega$, and let $P(K)$ denote the ordering parameter used in the sums. Suppose that $\operatorname{id}\notin\Omega$.

Rarity conjecture. For every $r=0,1,2,\dots$, there exists an integer $r'$ such that

$$\sum_{\substack{K\in\mathcal{S}\P(K)<x}}\mathbf{1}_{(\Omega,r)}(K)=o\left(\sum_{\substack{K\in\mathcal{S}\P(K)<x}}\mathbf{1}_{(\Omega,r')}(K)\right).$$

Moreover, for every $r=0,1,2,\dots$,

$$\sum_{\substack{K\in\mathcal{S}\P(K)<x}}\mathbf{1}_{(\Omega,r)}(K)=o\left(\sum_{\substack{K\in\mathcal{S}\P(K)<x}}1\right).$$

These assertions formalize the Malle–Bhargava heuristic that, among fields ordered by $P(K)$, fields with any fixed number of ramified primes whose inertia meets $\Omega$ are negligible compared with suitable families, and negligible compared with all fields. The source presents both assertions as conjectural; no resolution is supplied.