more abundant the monopoles, the lower the avalanche field. To obtain the data in Fig. 4, after the AQP and $t_{\rm w}$, the applied field was swept at a constant rate, 0.02 T.s$^{-1}$ up to 0.4 T. If the avalanche field $H_{\rm ava}(t_{\rm w})$, is defined as the field where the magnetization crosses 1 $\mu_{\rm B}$ per rare earth ion, (0.5 $\mu_{\rm B}$ for the $^{163}$Dy sample) then the difference in avalanche field $\Delta H_{\rm ava}= H_{\rm ava}(t_{\rm w}) - H_{\rm ava} (t_{\rm w}={\rm minimum})$ allows us to compare the spread of fields for all samples.

Fig. 4a and b show the experimental results for the isotopically enriched [Dy$_{2}$Ti$_{2}$O$_{7}$]{} samples at 80 mK, demonstrating a very clear pattern. In general the spread of $H_{\rm ava}(t_{\rm w})$ becomes larger, the larger the nuclear spin, showing again that the nuclear spins strongly enhance the monopole mobility. Thus, the $^{162}$Dy sample (no nuclear spin) shows negligible evolution of the position of the avalanche field. For $^{\rm nat}$[Dy$_{2}$Ti$_{2}$O$_{7}$]{}  (shown in Supplementary Fig. 4) the effect is small, while for the $^{163}$Dy sample (maximum nuclear spin) the effect of $t_{\rm w}$ can be clearly seen as a steady progression of $H_{\rm ava}(t_{\rm w})$ to higher 