The compact building strong transitivity characterization conjecture

Let $\Delta$ be an infinite thick irreducible compact spherical building of rank at least $2$. Strong transitivity characterization conjecture. The building $\Delta$ is the spherical building associated with a semisimple algebraic group over a non-discrete locally compact field if and only if $\Delta$ is strongly transitive. This extends the known characterization for connected compact buildings; the paper notes that strong transitivity could even be replaced by chamber transitivity, while the general disconnected case remains open.