The luminosity functions (LFs) of star cluster (SC) systems (i.e. the number of clusters per luminosity interval) are vital diagnostics to probe the conditions of SC formation. Early studies have revealed a clear dichotomy between old globular clusters and young clusters, with the former characterized by Gaussian-shaped LFs, and the latter following a power law. Recently, this view was challenged by studies of galaxy merger remnants and post-starburst galaxies. In this paper, we re-evaluate the young ( few hundreds of Myrs, with the majority few tens of Myrs) SC system in the ongoing spiral-spiral major merger system NGC 4038/39, the 'Antennae' galaxies. The Antennae galaxies represent a very active and complex star-forming environment, which hampers cluster selection and photometry as well as the determination of observational completeness fractions. A main issue of concern is the large number of bright young stars contained in most earlier studies, which we carefully exclude from our cluster sample by accurately determining the source sizes. The resulting LFs are fitted both with Gaussian and with power-law distributions, taking into account both the observational completeness fractions and the photometric errors, and compared using a likelihood-ratio test. The likelihood-ratio results are rigidly evaluated using Monte Carlo simulations. We perform a number of additional tests, for example, with subsets of the total sample, all confirming our main result: that a Gaussian distribution fits the observed LFs of clusters in this preferentially very young cluster system significantly better than a power-law distribution, at a (statistical) error probability of less than 0.5 per cent.