The compatible normal series conjecture for finite groups

Let $G$ and $H$ be finite groups. A compatible normal series for two groups is a pair of normal series whose corresponding factors are isomorphic in the same order. Two groups are compatible when they occur as regular permutation groups in a common finite group action, equivalently when they are compatible as abstract groups.

Compatible normal series conjecture. Two groups are compatible if and only if they have compatible normal series.

This conjecture would completely solve the compatibility problem for abstract groups. The paper's results construct compatible pairs from compatible normal series, while previous work established compatible normal series as a necessary condition in relevant cases; the converse in full generality remains open.