Let $ S $ be a finite group. Denote by $\mathcal{B}_0(S)$ the set of the subgroups $H$ of $S$ satisfying $|H:H'| > |K:K'|$ for every proper subgroup $K$ of $H$ ($H'$ denotes the drived subgroup of $H$), and let $\mathcal{K}(S)$ be the subgroup generated by the minimal elements of $\mathcal{B}_0(S)$ (the latter being ordered by inclusion). Is the following conjecture of Thompson ["A Replacement theorem for p-groups and a Conjecture" J. Algebra 13 (1969)] still open?

Conjecture. Let $G$ be a finite p-solvable group which doesn't involve $\mathrm{SL}_2(p)$ , and $S$ is a p-sylow of $G$. If $\mathrm{O}_{p'}(G) = 1$, then $\mathcal{K}(S) \lhd G$.