Derived line property trends with host galaxy properties. Left: integrated OH⁺ optical depth vs. dust temperature. The individual absorption components of MG J0414+0534 are indicated by smaller green points. Middle: OH⁺ emission line luminosity vs. L_FIR. Right: CO(9-8) emission line luminosity vs. L_FIR. In this panel, we include the nearby star-forming galaxy sample studied by Rosenberg et al. (2015), categorised into classes of lowest (class I) to highest (class III) CO excitation. We compare our QSO sample with other high-z QSOs (W0410−0913 Stanley et al. 2021 and SDSS J2310+1955 Shao et al. 2022) and DSFGs (HerBS-89a Berta et al. 2021, the Eyelash Indriolo et al. 2018, HLock-01 Scott et al. 2011 and the sample from Riechers et al. 2021) from the literature. In all panels, we colour the sources from Riechers et al. (2021) with blue-shifted OH⁺ absorption in black and colour the rest of the sample in grey.