  Let $N$ be a closed manifold and $U \subset T^*(N)$ a bounded domain in the
cotangent bundle of $N$, containing the zero-section. A conjecture due to
Viterbo asserts that the spectral metric for Lagrangian submanifolds that are
exact-isotopic to the zero-section is bounded. In this paper we establish an
upper bound on the spectral distance between two such Lagrangians $L_0, L_1$,
which depends linearly on the boundary depth of the Floer complexes of $(L_0,
F)$ and $(L_1, F)$, where $F$ is a fiber of the cotangent bundle.
