The elementary-abelian quotient conjecture for induced regular groups

Let $G$ be a finite group, let $Z(G)$ denote its center, and call $G$ induced regular when the induced non-centralizer graph on $G\setminus Z(G)$ is regular. A finite group is an elementary $p$-group if it is isomorphic to a direct product of copies of the cyclic group of order $p$ for some prime $p$. Elementary-abelian quotient conjecture. If $G$ is an induced regular group, then $G/Z(G)$ is an elementary $p$-group. This conjecture is motivated by the authors' observation that known induced regular groups include examples whose central quotient is nonabelian, while they found no example whose central quotient is not elementary; the preceding theorem reduces induced regular groups to an induced regular $p$-group times an abelian group. Whether every such quotient is elementary remains open in the supplied text.