Grids of the [CII]_158 μm/[CI]_369 μm intensity line ratio as a function of the strength of the radiation field (G₀ or F_X) and total hydrogen density (n_H) in the PDR (left panel) and XDR case (right panel). The adopted total hydrogen column density is N_H = 10²³ cm⁻². Model values are indicated by gray contours (linear scale), and constraints on the observed line ratios are superimposed. The dashed gold area indicates the [CII]_158 μm-to-[CI]_369 μm luminosity ratio measured in QSO PJ231−20 within its uncertainties, and 3σ lower limits on the other sources are reported (see the legend in the bottom left corner of the left panel). The [CII]_158 μm-to-[CI]_369 μm ratios in QSO PJ231−20 and its companion are scaled according to the estimated fraction of [CII]_158 μm arising from PDRs. The XDR case does not reproduce the observed ratios in any source, except for a small range of values corresponding to the lower limit measured in QSO PJ308−21. We also report the [CI]_369 μm-to-TIR and [CII]_158 μm-to-TIR luminosity ratio estimates in QSO PJ231−20 as filled gold and light brown areas, respectively. This provides additional constraints on the radiation field flux.