The early evolution of massive cluster progenitors is poorly understood. We investigate the fragmentation properties from 0.3 pc to 0.06 pc scales of a homogenous sample of infrared-quiet massive clumps within 4.5 kpc selected from the ATLASGAL survey. Using the ALMA 7 m array we detect compact dust continuum emission towards all targets and find that fragmentation, at these scales, is limited. The mass distribution of the fragments uncovers a large fraction of cores above 40 M⊙, corresponding to massive dense cores (MDCs) with masses up to ~400 M⊙. Seventy-seven percent of the clumps contain at most 3 MDCs per clump, and we also reveal single clumps/MDCs. The most massive cores are formed within the more massive clumps and a high concentration of mass on small scales reveals a high core formation efficiency. The mass of MDCs highly exceeds the local thermal Jeans mass, and we lack the observational evidence of a sufficiently high level of turbulence or strong enough magnetic fields to keep the most massive MDCs in equilibrium. If already collapsing, the observed fragmentation properties with a high core formation efficiency are consistent with the collapse setting in at parsec scales.