Paternain's real-analytic entropy conjecture for integrable geodesic flows

Let a closed manifold carry a geodesic flow that is analytically integrable, meaning integrable by real-analytic first integrals in the sense used in the source, and let $h_{\mathrm{top}}$ denote the topological entropy of the flow. Paternain's real-analytic entropy conjecture. If the geodesic flow is analytically integrable, then

$$h_{\mathrm{top}}=0.$$

The $C^{\infty}$ version of the entropy claim was disproved in the paper, while this real-analytic version is presented as the remaining conjecture; analytic integrability is also associated there with polynomial growth of the fundamental group.