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Two families of models compete to explain the formation of high-mass stars.The quasi-static models predict the existence of high-mass pre-stellar coressustained by a high degree of turbulence while competitive accretion modelspredict that high-mass proto-stellar cores evolve from low/intermediate massproto-stellar cores in dynamic environments. We present ALMA (1.4 mm continuumemission and $^{12}$CO emission line) and MOPRA (HCO$^{+}$, H$^{13}$CO$^{+}$and N$_2$H$^+$ molecular line emissions) observations of a sample of 9 starlessmassive dense cores (MDCs) discovered in a recent Herschel/HOBYS study thathave masses and sizes ($\sim$110 M$_\odot$ and $r\sim$0.1 pc, respectively)similar to the initial conditions used in the quasi-static models. The MOPRAmolecular line features show that 3 of the starless MDCs are subvirialized with$\alpha_{\rm vir}\sim$0.35, and 4 MDCs show sign of collapse. Our ALMAobservations, on the other hand, show very little fragmentation within the MDCswhereas the observations resolve the Jeans length ($\lambda_{\rmJeans}\sim$0.03 pc) and are sensitive to the Jeans mass (M$_{\rmJeans}\sim$0.65 M$_\odot$) in the 9 starless MDCs. Only two of the starlessMDCs host compact continuum sources, whose fluxes correspond to $<3$ M$_\odot$fragments. Therefore the mass reservoir of the MDCs has not yet been accretedonto compact objects, and most of the emission is filtered out by theinterferometer. These observations do not support the quasi-static models forhigh-mass star formation since no high-mass pre-stellar core is found inNGC6334. The competitive accretion models, on the other hand, predict a levelof fragmentation much higher than what we observe.

Lack of high-mass prestellar cores in the starless MDCs of NGC6334