We assess the performance of a group of exchange-correlation functionals for predicting the secondary structure of peptide chains, up to a new many-body dispersion corrected hybrid density functional, dubbed PBE0+MBD* by its original authors. For the purpose of validation, we first compare to published, high-level benchmark conformational energy hierarchies (coupled cluster at the singles, doubles, and perturbative triples level, CCSD(T)) for 73 conformers of small three-residue peptides, establishing that the van der Wash corrected PBE0 functional yields an average error of only similar to 20 meV (similar to 0.5 kcal/mol). This compares to similar to 40-50 meV for non-dispersion corrected PBE0 and 40-100 meV for different empirical force fields (estimated for the alanine tetrapeptide). For longer peptide chains that form a secondary structure, CCSD(T) level benchmark data are currently unaffordable. We thus turn to the experimentally well studied Ac-Phe-Ala(5)-LysH(+) peptide, for which four closely competing conformers were established by infrared spectroscopy. For comparison, an exhaustive theoretical conformational space exploration yields at least 11 competing low energy minima. We show that (i) the many-body dispersion correction, (ii) the hybrid functional nature of PBE0+MBD*, and (iii) zero-point corrections are needed to reveal the four experimentally observed structures as the minima that would be populated at low temperature.