Proof of the index conjecture in Hofer geometry

Let $\gamma$ be a non-degenerate Ustilovsky geodesic in $Ham(M,\omega )$ generated by $H$. We give a simple proof of a generalization of the conjecture stated in \cite{virtmorse}, relating the Morse index of $\gamma$, as a critical point of the Hofer length functional, with the Conley Zehnder index of the extremizers of $H$, considered as periodic orbits.