Solubilizer Conjecture A.16

Conjecture $A.16$ asks whether a proper solubilizer in a nonsolvable finite group has metabelian intersection with its normalizer. It is listed in a conjecture-mining paper, which gives no proof or counterexample.

Reported counterexample (date not stated)

A report claims that $G=A_5\times S_4$ has a proper self-normalizing solubilizer $D_{10}\times S_4$, whose derived length is $3$ rather than at most $2$; this would refute Conjecture $A.16$. The claim is supported by finite-group computation and independent recomputation, but remains unverified.

Current status (as of July 2026): A specific counterexample is claimed for $A_5\times S_4$, but no independently refereed proof or verification has been found; absent that, the conjecture's status remains unsettled.