Universal Liouville and noncommutative integrability conjecture for magnetic flows on spheres

Let $m$ be the mass of a material point moving on the sphere $S^{n-1}\subset\mathbb{R}^n$, let the point be placed in a constant homogeneous magnetic field described by the parameters $\kappa$, and consider the restriction of the corresponding magnetic system to $S^{n-1}$. Universal integrability conjecture. Motion on $S^{n-1}$ is completely integrable in the Liouville or in the noncommutative sense for any $n$ and $\kappa$. The paper establishes complete integrability in dimensions up to six and noncommutative integrability in higher dimensions under reductions, while the asserted unrestricted statement for arbitrary $n$ and $\kappa$ is not established.