The toric integrability conjecture for metrics on the torus

Let $g$ be a Riemannian metric on the torus $\mathbb{R}^n/\mathbb{Z}^n$, and suppose that its geodesic flow is toric integrable. Toric integrability conjecture. Then $g$ is flat. This is a geometric formulation related to the Hopf conjecture; the source states that it was subsequently proved by E. Lerman and N. Shirokova.