Correlation between far-IR luminosity and HCN/HCO⁺(3–2) and (4–3) luminosities. Solid/dashed orange lines indicate the empirical trends from this work and Zhang et al. (2014) with ±1σ scatter; dotted orange lines indicate the empirical trend from Nishimura et al. (2024). Individual data points show z = 0 galaxy-averaged (Greve et al. 2009; Papadopoulos et al. 2014; Zhang et al. 2014; Nishimura et al. 2024) and resolved observations (Tan et al. 2018), and high-z detections in individual galaxies (Riechers et al. 2011; Cañameras et al. 2021, black, solid) and this work (blue). We also show results from spectral stacks (Reuter et al. 2023; Hagimoto et al. 2023). Where appropriate, we corrected for the lensing magnification. We also plot error bars on high-z measurements; these are typically smaller than the symbol size.