Bounds on the Lagrangian spectral metric in cotangent bundles

Let $N$ be a closed manifold and $U \subset T^\*(N)$ bounded domain in the cotangent bundle of $N$, containing zero-section. A conjecture due to Viterbo asserts that spectral metric for Lagrangian submanifolds $U$ are exact-isotopic zero-section is bounded. In this paper we establish an upper bound on distance between two such Lagrangians $L\_0, L\_1$, which depends linearly boundary depth Floer complexes $(L\_0, F)$ $(L\_1, F)$, where $F$ fiber bundle.