Viterbo conjecture for Zoll symmetric spaces

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,RPn,CPn,HPn,n >= 1. We discuss generalizations and give applications, in particular to C0 symplectic topology. Our key method consists in a quantitative deformation argument for Floer persistence modules that allows to excise a divisor.