Understanding and tuning the ligand shell composition in colloidal halide perovskite nanocrystals (NCs) has been done systematically only for Pb-based perovskites, while much less is known on the surface of Pb-free perovskite systems. Here, we reveal the ligand shell architecture of Bi-doped Cs2Ag1-xNaxInCl6NCs via nuclear magnetic resonance analysis. This material, in its bulk form, was found to have a photoluminescence quantum yield (PLQY) as high as 86%, a record value for halide double perovskites. Our results show that both amines and carboxylic acids are present and homogeneously distributed over the surface of the NCs. Notably, even for an optimized surface ligand coating, achieved by combining dodecanoic acid and decylamine, a maximum PLQY value of only 37% is reached, with no further improvements observed when exploiting post-synthesis ligand exchange procedures (involving Cs-oleate, different ammonium halides, thiocyanates and sulfonic acids). Our density functional theory calculations indicate that, even with the best ligands combination, a small fraction of unpassivated surface sites, namely undercoordinated Cl ions, is sufficient to create deep trap states, opposite to the case of Pb-based perovskites that exhibit much higher defect tolerance. This was corroborated by our transient absorption measurements, which showed that an ultrafast trapping of holes (most likely mediated by surface Cl-trap states) competes with their localization at the AgCl6octahedra, from where, instead, they can undergo an optically active recombination yielding the observed PL emission. Our results highlight that alternative surface passivation strategies should be devised to further optimize the PLQY of double perovskite NCs, which might include their incorporation inside inorganic shells.