Some formation scenarios that have been put forward to explain multiple populations within Globular Clusters (GCs) require that the young massive cluster have large reservoirs of cold gas within them, which is necessary to form future generations of stars. In this paper we use deep observations taken with Atacama Large Millimeter/sub-millimeter Array (ALMA) to assess the amount of molecular gas within 3 young (50-200 Myr) massive (~10^6 Msun) clusters in the Antennae galaxies. No significant CO(3--2) emission was found associated with any of the three clusters. We place upper limits for the molecular gas within these clusters of ~1x10^5 Msun (or <9 % of the current stellar mass). We briefly review different scenarios that propose multiple episodes of star formation and discuss some of their assumptions and implications. Our results are in tension with the predictions of GC formation scenarios that expect large reservoirs of cool gas within young massive clusters at these ages.
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