Ratio of the predicted flux from our BH population and the observed limits in the soft band (0.3–2 keV; shown by the color bar). The solid red line denotes the curve where the ratio of the predicted and observed flux is equal to one. The y-axis shows the metallicities from solar to Z = 0.01 Z_⊙. The x-axis is the average column density N_ave in units of cm⁻². We adopted k = 0.3 as an example of an extreme case of high activity (top panel) and a more conservative value k = 2 × 10⁻³ (bottom panel).