Coset-fibration conjecture for totally geodesic foliations in compact Lie groups

Let $G$ be a simple compact Lie group with a bi-invariant metric, and let $\pi:G\rightarrow B$ be a Riemannian submersion with connected totally geodesic fibers. Coset-fibration conjecture. The map $\pi$ is a left or right coset fibration; that is, $B=G/H$ or $B=H\backslash G$ for some subgroup $H\subset G$, and $\pi$ is the quotient map. This conjecture concerns the classification of Riemannian submersions from compact Lie groups with bi-invariant metrics. It was proved under the additional hypothesis that the fiber through the identity contains a maximal torus of $G$, while the unrestricted statement is resolved according to the supplied status evidence.