Recovery of the theoretical X_off distribution using the Δ_X − O − X_off relation. The application of Eq. (8) on the offset measured in Magneticum and TNG is shown in red and violet, respectively. The shaded areas contain the 1σ uncertainty on the model of the Δ_X − O − X_off relation. The blue (orange) dashed line refer to the direct measure of X_off in the DMO counterpart of Magneticum (TNG). We find an excellent agreement between the distribution of the true X_off and the prediction of Eq. (8).