CO luminosity plotted against the linewidth (ΔV) of the CO emission line for the sources of the Pilot Program reported in this paper (large red squares), compared to high-z lensed and unlensed galaxies, as well as local ultra-luminous infrared galaxies (ULIRGs) from the literature with corresponding symbols as indicated in the figure. No correction for amplification was applied to the CO luminosities. The sources of the Pilot Program are identified by the number of the HerBS catalogue (Table 1 and Bakx et al. 2018). Individual galaxies are also identified, namely two binary hyper-luminous galaxies (1) HATLAS J084933 at z = 2.41 (Ivison et al. 2013), where the two main components, W and T, are separately labeled; (2) HXMM01 (H-A and H-B) at z = 2.308 (Fu et al. 2013); other well-known sources, both lensed (IRASF10214, Eyelash, Cloverleaf, APM08279 and the Cosmic Eyebrow) and unlensed (BR1202N and S, and BR1335) from Carilli & Walter (2013, and references therein) and Dannerbauer et al. (2019). We note that the sources of Harris et al. (2012) are the ones reported in Table 1 of that paper, except for HATLAS J084933 where we used the follow-up measurements of Ivison et al. (2013). With the exception of the sources of this paper and the sources selected from Yang et al. (2017) and Bothwell et al. (2013), all the other sources used in this plot have been measured in ¹²CO(1−0) or, in some cases, in ¹²CO(2−1). Corrections for excitation were applied for sources for which only higher CO transitions are available (see Sect. 4.5). The dashed line shows the best-fitting power-law fit derived from the data for the unlensed SMGs, (e.g., Bothwell et al. 2013; Zavala et al. 2015).