Viterbo's spectral bound conjecture for cotangent bundles

Let $(M,g)$) be a closed Riemannian manifold. Define

$$DT^*M\coloneqq \{(x,\xi)\in T^*M\mid \|\xi\|_g\le 1\}$$

and let $0_M$ denote the image of the zero section of $T^*M$. For a compactly supported Hamiltonian diffeomorphism $\phi\in\operatorname{Ham}_c(T^*M)$, let $\gamma(\phi(0_M))$ denote the spectral invariant of the image of the zero section. Viterbo conjecture. There exists a constant $R>0$ such that, if $\phi\in\operatorname{Ham}_c(T^*M)$ satisfies

$$\phi(0_M)\subset DT^*M,$$

then

$$\gamma(\phi(0_M))<R.$$

This conjecture is a spectral bound for images of the zero section contained in the unit cotangent disk bundle; the source notes that it was first conjectured by Viterbo in the special case $M=T^n$.