described in @milone2017 [see their Section 3]. Finally, we derived the kernel-density distributions of stars in the verticalized I vs $\Delta$($U-I$) CMDs (red lines in the right panels of Figure \[fig:cmds\]) and identified by eye the groups of Fe-poor (orange dots) and Fe-rich (cyan triangles) RGB stars, which are located on the left and right side of the vertical dashed line, respectively. In the case of $\omega\,$Centauri we adopted an intermediate step, before identifying Fe-poor and Fe-rich stars in the *HST* inner field. Specifically, to ensure consistency between the two fields and data sets, we converted the $m_{F336W}$ and $m_{F814W}$ magnitudes into $U$ and $I$ magnitudes. The same process was redundant in the simpler case of M22, as revealed by the right panels of Figure \[fig:cmds\].\
To verify the impact of the adopted selection of $\omega$Centauri stars with different metallicities on the conclusions of the paper, we adopted two additional procedures (II and III). Procedure II consists of excluding stars with $U-I$ colors within $\pm$0.03 mag from vertical dashed line from the metal-poor and metal-rich sample defined above. Procedure III is based on the Monte-Carlo method for selection of metal-rich and metal-poor stars. We fitted the $\Delta$($U-I$) distributions of stars 