The intersection-multiplicity conjecture for field families

Let $k$ be a number field, let $G$ be a finite group, let $\pi\colon G\to S_n$ be a transitive faithful permutation representation with $n\geq2$, and let $N\unlhd G$ be normal. Define $\mathfrak{F}_k^{G,\pi}(Q)$ as the fields whose associated subextension $K^\pi$ has discriminant at most $Q$, and define

$$\mathfrak{m}_k^{G,N,\pi}(Q):=\max_{K_1\in\mathfrak{F}_k^{G,\pi}(Q)}\left|\{K_2\in\mathfrak{F}_k^{G,\pi}(Q):K_1\cap K_2\neq K_1^N\cap K_2^N\}\right|.$$

For $N'\unlhd G$, set

$$a_\pi(G,N'):=\min\{n-\#\operatorname{Orb}_\pi(g):g\in N',\ g\neq\operatorname{id}\},$$

and define

$$m_\pi(G,N):=\max_{\substack{N'\unlhd G\N\not\subseteq N'}}a_\pi(G,N')^{-1},$$

with $m_\pi(G,N)=0$ if there is no nontrivial such $N'$. The intersection-multiplicity conjecture. As $Q\to\infty$,

$$Q^{m_\pi(G,N)}\ll_{k,G}\mathfrak{m}_k^{G,N,\pi}(Q)\ll_{k,G,\varepsilon}Q^{m_\pi(G,N)+\varepsilon}.$$

This conjecture refines the counting heuristics behind Malle's conjecture and would control the subfield or intersection obstruction in the paper's average estimates. Its general validity remains open.