with the aim to determine the effect of nuclear spins on the monopole dynamics. Varying the wait time deep in the frozen regime allowed us to gauge the spontaneous evolution of the zero-field monopole density as a function of time: that is, if monopoles recombine in a time $t_{\rm w}$, then the observed monopole current will be smaller, the longer the wait time. Two separate experiments were designed to study these effects. In the first experiment (Fig. 3) after waiting we applied a constant field and measured the magnetization $M$ as a function of time. In the second experiment (Fig. 4) we investigated the effect of wait time on the magnetothermal avalanches [@Slobinsky; @Jackson; @Krey2012] that occur on ramping the field to high values. Both of these allowed access to the magnetic current density $J_{\rm m} = dM / dt$. Full details of the experimental conditions are given in Supplementary Note 1 and Supplementary Fig. 2.

The monopole current is controlled by multiple factors. In the simplest model [@Ryzhkin] there are three of these: the monopole density $n$, the monopole mobility $u$ (related to the spin tunnelling rate) and the bulk susceptibility $\chi$. Thus $J_{\rm m} = dM/dt = \nu 