On Gorenstein ${\mathbb{Q}}_p$-rational threefolds and fourfolds

We prove that for $n\le 4$ and $p>5$, quasi–Gorenstein $F$–pure and ${\mathbb{Q}}_p$–rational $n$–fold singularities are canonical. This is analogous to the usual fact that rational Gorenstein singularities are canonical. The proof is based on a careful analysis of the dual complex of a dlt modification of a log canonical singularity. The result for $n=4$ is contingent upon the existence of log resolutions.