Bounded Lagrangians are spectrally bounded

Let $(M,\omega)$ be a symplectic manifold and let $L$ be an exact Lagrangian. For any Hamiltonianly isotopic exact Lagrangian $L_1$ such that $\gamma(L,L_1)$ is sufficiently small, consider Hamiltonian isotopies $\varphi^t$ preserving $L$. Bounded Lagrangians are spectrally bounded. There exists a constant $c$ depending on the pair $(M,L)$ such that one can choose such an isotopy with

$$\gamma(L_1,\varphi^1(L_1))\geq c\gamma(L,L_1).$$

This conjecture asserts a uniform lower bound on the spectral displacement produced by an isotopy preserving the reference Lagrangian, proportional to the distance from $L$ to $L_1$. The supplied text gives no resolution status.