The linking–transporter extension conjecture for almost simple fusion systems

Let $\calf$ be an almost simple fusion system over a $p$-group $S$. More generally, assume only that $O_p(\calf)=1$. Set $\cale=F^*(\calf)$; thus $\cale$ is simple, or a product of simple fusion systems, depending on the assumption. A linking system is a category associated to a fusion system, and a transporter system is the corresponding extension structure. Linking–transporter extension conjecture. There exist a linking system $\call$ associated to $\cale$ and a transporter system $\calt$ associated to $\calf$ such that $\call\nsg\calt$. The conjecture concerns constructing compatible categorical extensions of products of simple fusion systems and is proposed as a foundation for understanding such extensions; the source provides no resolution.