The Main Conjecture on AIM loops

Let $Q$ be an AIM loop, meaning that its inner mapping group $\operatorname{Inn}(Q)$ is abelian. Let $N(Q)$ denote the nucleus of $Q$ and $Z(Q)$ its center; the factor loop $Q/N(Q)$ is understood to be defined, so $N(Q)$ is normal in $Q$.

The Main Conjecture. The quotient $Q/N(Q)$ is an abelian group and $Q/Z(Q)$ is a group. In particular, $Q$ is nilpotent of class at most $3$.

The conjecture is motivated by the structure of known loops of Csörgő type and is stronger than the asserted nilpotency-class bound. It does not impose a finiteness assumption; the source notes that it might hold for all finite AIM loops while admitting an infinite counterexample. The earlier claim that every AIM loop has nilpotency class at most $2$ was disproved, so this structural replacement remains open.