The conjecture that saturated finite groups are abelian

Let $G$ be a finite group. Call $G$ saturated if every saturated $G$-transfer system comes from a linear isometries operad.

Abelianness conjecture for saturated groups. If $G$ is a saturated finite group, then $G$ is abelian.

The question concerns which finite groups have the property that all saturated transfer systems arise from linear isometries operads. Many abelian cases are known, while the paper proves that several infinite families of non-abelian groups are not saturated; the existence of any saturated non-abelian group remains open.