Context : The formation of high-mass stars remains unknown in many aspects .
Two families of models compete to explain the formation of high-mass stars .
On the one hand , quasi-static models predict the existence of high-mass pre-stellar cores sustained by a high degree of turbulence .
On the other hand competitive accretion models predict that high-mass proto-stellar cores evolve from low/intermediate mass proto-stellar cores in dynamic environments .

Aims : The aim of the present work is to bring observational constraints at the scale of high-mass cores ( \sim 0.03 pc ) .

Methods : We targeted with ALMA and MOPRA a sample of 9 starless massive dense cores ( MDCs ) discovered in a recent Herschel/HOBYS study .
Their mass and size ( \sim 110 M _ { \odot } and r =0.1 pc , respectively ) are similar to the initial conditions used in the quasi-static family of models explaining for the formation of high-mass stars .
We present ALMA 1.4mm continuum observations that resolve the Jeans length ( \lambda _ { Jeans } \sim 0.03 pc ) and that are sensitive to the Jeans mass ( M _ { Jeans } \sim 0.65 M _ { \odot } ) in the 9 starless MDCs , together with ALMA- ^ { 12 } CO ( 2-1 ) emission line observations .
We also present HCO ^ { + } ( 1-0 ) , H ^ { 13 } CO ^ { + } ( 1-0 ) and N _ { 2 } H ^ { + } ( 1-0 ) molecular lines from the MOPRA telescope for 8 of the 9 MDCs .

Results : The 9 starless MDCs have the mass reservoir to form high-mass stars according to the criteria by [ ] .
Three of the starless MDCs are subvirialized with \alpha _ { vir } \sim 0.35 , and 4 MDCs show sign of collapse from their molecular emission lines .
ALMA observations show very little fragmentation within the MDCs .
Only two of the starless MDCs host compact continuum sources , whose fluxes correspond to < 3 M _ { \odot } fragments .
Therefore the mass reservoir of the MDCs has not yet been accreted onto compact objects , and most of the emission is filtered out by the interferometer .

Conclusions : These observations do not support the quasi-static models for high-mass star formation since no high-mass pre-stellar core is found in NGC6334 .
The competitive accretion models , on the other hand , predict a level of fragmentation much higher than what we observe .