The geometrically bounded implies spectrally bounded conjecture

Let $N$ be a closed Riemannian manifold, and let

$$DT^*N=\left\{(q,p)\in T^*N\mid |p|_g\leq 1\right\}.$$

An exact Lagrangian $L$ is geometrically bounded when it is contained in $DT^*N$. Geometrically bounded implies spectrally bounded conjecture. There exists a constant $C_N$ such that every exact Lagrangian $L\subset DT^*N$ satisfies

$$\gamma(L)\leq C_N.$$

The source recalls that this conjecture was proved in some special cases, while its general form is presented as an outstanding conjecture.