COALAS II. Extended molecular gas reservoirs are common in a distant, forming galaxy cluster

This paper presents the results of 475 hours of interferometric observations with the Australia Telescope Compact Array towards the Spiderweb protocluster at $z=2.16$. We search for large, extended molecular gas reservoirs among 46 previously detected CO(1-0) emitters, employing a customised method we developed. Based on the CO emission images and position-velocity diagrams, as well as the ranking of sources using a binary weighting of six different criteria, we have identified 14 robust and 7 tentative candidates that exhibit large extended molecular gas reservoirs. These extended reservoirs are defined as having sizes greater than 40 kpc or super-galactic scale. This result suggests a high frequency of extended gas reservoirs, comprising at least $30 \%$ of our CO-selected sample. An environmental study of the candidates is carried out based on N-th nearest neighbour and we find that the large molecular gas reservoirs tend to exist in denser regions. The spatial distribution of our candidates is mainly centred on the core region of the Spiderweb protocluster. The performance and adaptability of our method are discussed. We found 13 (potentially) extended gas reservoirs located in nine galaxy (proto)clusters from the literature. We noticed that large extended molecular gas reservoirs surrounding (normal) star-forming galaxies in protoclusters are rare. This may be attributable to the lack of observations low-J CO transitions and the lack of quantitative analyses of molecular gas morphologies. The large gas reservoirs in the Spiderweb protocluster are a potential source of the intracluster medium seen in low redshift Virgo- or Coma-like galaxy clusters.