Galactic globular clusters (GCs) are known to host multiple stellar populations: a first generation (FG) with a chemical pattern typical of halo field stars and a second generation (SG) enriched in Na and Al and depleted in O and Mg. Both stellar generations are found at different evolutionary stages (e.g., the main-sequence turnoff, the subgiant branch, and the red giant branch (RGB)). The non detection of SG asymptotic giant branch (AGB) stars in several metal-poor ([Fe/H] <‑1) GCs suggests that not all SG stars ascend the AGB phase, and that failed AGB stars may be very common in metal-poor GCs. This observation represents a serious problem for stellar evolution and GC formation/evolution theories. We report fourteen SG-AGB stars in four metal-poor GCs (M13, M5, M3, and M2) with different observational properties: horizontal branch (HB) morphology, metallicity, and age. By combining the H-band Al abundances obtained by the Apache Point Observatory Galactic Evolution Experiment survey with ground-based optical photometry, we identify SG Al-rich AGB stars in these four GCs and show that Al-rich RGB/AGB GC stars should be Na-rich. Our observations provide strong support for present, standard stellar models, i.e., without including a strong mass-loss efficiency, for low-mass HB stars. In fact, current empirical evidence is in agreement with the predicted distribution of FG and and SG stars during the He-burning stages based on these standard stellar models.