Young massive clusters (YMCs) are the most compact, high-mass stellar systems still forming at the present day. The precursor clouds to such systems are, however, rare due to their large initial gas mass reservoirs and rapid dispersal timescales due to stellar feedback. Nonetheless, unlike their high-z counterparts, these precursors are resolvable down to the sites of individually forming stars, and hence represent the ideal environments in which to test the current theories of star and cluster formation. Using high angular resolution (1$^{primeprime}$ / 0.05pc) and sensitivity ALMA observations of two YMC progenitor clouds in the Galactic Centre, we have identified a suite of molecular line transitions -- e.g. c-C$_{3}$H$_{2} $($7-6$) -- that are believed to be optically thin, and reliably trace the gas structure in the highest density gas on star-forming core scales. We conduct a virial analysis of the identified core and proto-cluster regions, and show that half of the cores (5/10) and both proto-clusters are unstable to gravitational collapse. This is the first kinematic evidence of global gravitational collapse in YMC precursor clouds at such an early evolutionary stage. The implications are that if these clouds are to form YMCs, then they likely do so via the conveyor-belt mode, whereby stars continually form within dispersed dense gas cores as the cloud undergoes global gravitational collapse. The concurrent contraction of both the cluster-scale gas and embedded (proto)stars ultimately leads to the high (proto)stellar density in YMCs.
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