Equality of the Hofer and virtual indices for Hamiltonian circle actions

Let $(M,\omega)$ be a closed symplectic manifold, and let $\gamma$ be a Hamiltonian circle action generated by a Morse Hamiltonian. The Hofer index $I^{H}(\gamma)$ is the maximal dimension of a subspace of $T_{\gamma}\Omega\operatorname{Ham}(M,\omega)$ on which the Hessian of the positive Hofer length is negative definite, while $I^{\mathrm{virt}}(\gamma)$ denotes the virtual index. Equality of the Hofer and virtual indices.

$$I^{H}(\gamma)=I^{\mathrm{virt}}(\gamma).$$

In the supplied text this statement is presented as a conjecture, although the surrounding discussion also states the equality as a theorem under the preceding hypotheses; its resolution should therefore be checked against the paper’s final published version.