Mishchenko–Fomenko conjecture for homogeneous geodesic flows

Let $G$ be a compact Lie group, let $H$ be a subgroup, and let $\mathfrak v$ denote the corresponding tangent-space representation used to form the invariant polynomial algebra $\mathbb{R}[\mathfrak v]^H$. Call $(G,H)$ an integrable pair if there exists a complete commutative subalgebra $\mathcal B\subset\mathbb{R}[\mathfrak v]^H$. The normal metric on $G/H$ has a geodesic flow that is non-commutatively integrable. Mishchenko–Fomenko conjecture for homogeneous geodesic flows. All pairs $(G,H)$ are integrable. The assertion would imply that every such normal homogeneous geodesic flow admits a complete commutative algebra of analytic first integrals polynomial in velocities; the source presents it as the homogeneous-space formulation of the Mishchenko–Fomenko conjecture and gives no resolution status.