  We prove a conjecture of Viterbo from 2007 on the existence of a uniform
bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero
section in unit cotangent disk bundles, for bases given by compact rank one
symmetric spaces $S^n, \mathbb{R} P^n, \mathbb{C} P^n, \mathbb{H} P^n,$ $n\geq
1.$ We discuss generalizations and give applications, in particular to $C^0$
symplectic topology. Our key methods, which are of independent interest,
consist of a reinterpretation of the spectral norm via the asymptotic behavior
of a family of cones of filtered morphisms, and a quantitative deformation
argument for Floer persistence modules, that allows to excise a divisor.
