Context. The different theoretical models concerning the formation of high-mass stars make distinct predictions regarding their progenitors, i.e. the high-mass prestellar cores. However, so far no conclusive observation of such objects has been made. Aims. We aim to study the very early stages of high-mass star formation in two infrared-dark, massive clumps, to identify the core population that they harbour. Methods. We obtained ALMA observations of continuum emission at 0.8mm and of the ortho-$rm H_2D^+$ transition at 372GHz towards the two clumps. We use the SCIMES algorithm to identify cores in the position-position-velocity space, finding 16 cores. We model their observed spectra in the LTE approximation, deriving the centroid velocity, linewidth, and column density maps. We also study the correlation between the continuum and molecular data, which in general do not present the same structure. Results. We report for the first time the detection of ortho-$rm H_2D^+$ in high-mass star-forming regions performed with an interferometer. The molecular emission shows narrow and subsonic lines, suggesting that locally the temperature of the gas is less than 10K. From the continuum emission we estimate the cores total masses, and compare them with the respective virial masses. We also compute the volume density values, which are found to be higher than $10^{6}, rm cm^{-3}$. Conclusions. Our data confirm that ortho-$rm H_2D^+$ is an ideal tracer of cold and dense gas. Interestingly, almost all the $rm H_2D^+$-identified cores are less massive than 13M_sun , with the exception of one core in AG354. Furthermore, most of them are subvirial and larger than their Jeans masses. These results are difficult to explain in the context of the turbulent accretion models, which predict massive and virialised prestellar cores.
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