Non-Lyapunov annealed decay for 1d Anderson eigenfunctions

In [10] Jitomirskaya, Krüger and Liu analysed the dynamical decay in expectation for the super-critical almost-Mathieu operator in function of the coupling parameter , showing that it is equal to the Lyapunov exponent of its transfer matrix cocycle, and asked whether the same is true for the 1d Anderson model. We show that this is essentially never true when the disorder parameter is sufficiently large.