With the operation of LOFAR, a great opportunity exists to shed light on a problem of some cosmological significance. Diffuse radio synchrotron emission not associated to any obvious discrete sources as well as Faraday rotation in clusters of galaxies both indicate that the intergalactic or intracluster medium (IGM, ICM) is pervaded by a weak magnetic field, along with a population of relativistic particles. Both, particles and fields must have been injected into the IGM either by Active Galactic Nuclei (AGN) or by normal star-forming galaxies. Excellent candidates for the latter are starburst dwarf galaxies, which in the framework of hierarchical structure formation must have been around in large numbers. If this is true, one should be able to detect extended synchrotron halos of formerly highly relativistic particles around local starburst or post-starburst dwarf galaxies. With LOFAR, one should easily find these out to the Coma Cluster and beyond.