The eigenvalues of the Sinyukov mapping for geodesically equivalent metrics are globally ordered

Suppose all geodesics of two Riemannian metrics $g$ and $\overline g$ defined on a (connected, geodesically complete) manifold $M^n$ coincide. At each point $x\in M^n$, consider the common eigenvalues $\rho_1,\rho_2,\dots,\rho_n$ of the two metrics (we assume that $\rho_1\geqslant\rho_2\geqslant\dots\geqslant\rho_n$)) and the numbers
$$ \lambda_i=(\rho_1\rho_2\dotsb\rho_n)^{1/(n+1)}\frac1{\rho_i}. $$
. We show that the numbers $\lambda_i$ are ordered over the entire manifold: for any two points $x$ and $y$ in M the number $\lambda_k(x)$ is not greater than $\lambda_{k+1}(y)$. If $\lambda_k(x)=\lambda_{k+1}(y)$, then there is a point $z\in M^n$ such that $\lambda_k(z)=\lambda_{k+1}(z)$. If the manifold is closed and all the common eigenvalues of the metrics are pairwise distinct at each point, then the manifold can be covered by the torus.