Coexistence of measures with simple Lyapunov spectrum for fiber-bunched cocycles

We prove that if a Holder continuous fiber-bunched cocycle  over an invertible hyperbolic transitive shift Σ satisfies an appropriate strong irreducibility condition on Grassmannians, then Σ admits an ergodic measure μ with full support and product structure with simple Lyapunov spectrum if and only if any other ergodic measure with full support and product structure also has simple Lyapunov spectrum.