Survey design: the redshifts and total IR luminosities distributions of the primary CO (5 − 4) sample of our campaign. Gray crosses mark uncertain sources due to poor optical redshifts (Flag = −1). Gray open squares indicate uncertain upper limits on every covered line with trustable z_spec, opt (Flag = 0). Light blue, blue, and navy blue filled circles indicate reliable sources with usable information about one, two, or more than two transitions from this work (Flag ≥ 0.5). Red open circles mark the objects already presented in V18 and V20. Galaxies without an updated far-IR modeling based on the photometry in Jin et al. (2018) are not shown. The flagging classification and the definition of reliable and uncertain data are described in Sect. 3.2.2.