The inner mapping group conjecture for loops

Let $Q$ be a loop, and let $\operatorname{Inn}(Q)$ denote its inner mapping group. A loop is centrally nilpotent if it has a central series; its nilpotence class is the length of the shortest such series. Inner mapping group conjecture. If $\operatorname{Inn}(Q)$ is abelian, then $Q$ is centrally nilpotent of class at most $3$. This is an explicitly stated open problem concerning the implications among nilpotence and solvability properties of loops. The surrounding discussion indicates that several related implications are known, sometimes only for finite loops, while this implication remains open.