We consider a doubly-rescaled zero-range process with jump rate $g(k)=k^\alpha, \alpha>1$, with scaling parameters $\chi_N\to 0, N\to \infty$, as a microscopic model for the porous medium equation. As a result of the superlinear jump rate, new ingredients are needed in addition to the Kipnis-Landim framework, of which the most interesting is an integrability estimate: Even if one can prove rapid equilibration on macroscopically small boxes, the superexponential estimate could fail due to configurations in which a vanishing proportion of mass produces a nonvanishing contribution to the $L^\alpha_{t,x}$ norm. In order to rule this out, we show that the realisations of the particle system enjoy pathwise regularity estimates with superexponentially high probability across suitably chosen scales, which can be used in a multiscale argument to obtain the necessary integrability. Joint work with Benjamin Gess (TU Berlin / Max-Planck Institute for Mathematics in the Sciences)