Kozlov and Fomenko's nonexistence conjecture for higher-degree integrals on the two-sphere

On the two-dimensional sphere, consider Riemannian metrics whose geodesic flows are integrable by means of an integral of degree $n>4$ in the momenta. Kozlov and Fomenko's conjecture. There are no such metrics that do not also admit integrals of degree $c=4$. This conjecture concerns the possible degrees of polynomial integrals of geodesic flows and motivates the construction of superintegrable systems with higher-degree integrals; its resolution status is not specified in the source.