at $t \approx 10^8$ the $\langle Y_2 \rangle_{p_0}$ do not differ strongly from each other, so that the hints onto the dynamical localization are present in each single realization of the random traps. Especially at the onset of the plateau at $t \approx 10^2$ all different $\langle Y_2 \rangle_{p_0}$ appear to follow the same pattern, independent on what equilibrium value of $\langle Y_2\rangle_\tau$ would be attained. Fig. \[fig:Y2p0av\] shows that, even though the temporal evolution of the sample $\langle Y_2\rangle_{p_0}$ fluctuates around the disorder average $\langle Y_2\rangle_{p_0, \tau}$, the effect of the dynamical localization is not an exclusive property of $\langle Y_2\rangle_{p_0, \tau}$ but does already appear in the average over initial conditions and thermal histories.

The analytical calculation of $Y_2$ in the non-equilibrium case pertinent to dynamical localization turned out to be quite difficult. Bertin and Bouchaud [@Bertin2002] conjectured $$\begin{aligned}
  \langle Y_2 \rangle_\tau^{(dyn)} &= \frac{2}{3}~\langle Y_2 \rangle_\tau^{(eq)},
  \label{equ:BBconjecture}\end{aligned}$$ using a simple approximation and presented simulations with infinite-time extrapolation that confirmed their conjecture. We find it worth mentioning that according to their simulations [@Bertin2002 Fig. 5] as well as in our attempts to reproduce their figure, the simulated data tend to be smaller than Eq. (\[equ:BBconjecture\]) predicts, whereas it is 