Lucchini–Detomi's monolithicity conjecture for sigma-elementary groups

Let $G$ be a group with finite covering number

\sigma(G)$, where

sigma(G)$is the smallest number of proper subgroups needed to cover$G$. Call$G$ **

\sigma$-elementary** if

$\sigma(G)<$sigma(G/N)

for every non-trivial normal subgroup $N$ of $G$. A group is **monolithic** if it has a unique minimal normal subgroup. **Lucchini–Detomi's conjecture.** Every non-abelian

sigma$-elementary group is monolithic. This conjecture concerns the structure of groups that are minimal, with respect to quotienting by non-trivial normal subgroups, among groups with a given covering number. The source gives no resolution, so the conjecture is recorded as open.