Scramjet engines are high-speed airbreathing propulsion systems that do not require rotating elements to compress the air inlet stream. It is compressed dynamically through a supersonic intake system that is integrated in the forebody, thus leading to the required temperature and pressure levels for combustion to proceed within the combustor length. In such engines, the combustion chamber is crossed by a supersonic flow, which limits the time available to inject fuel, to mix it with oxidizer, to ignite the resulting mixture, and to reach complete combustion. Residence times can be increased thanks to cavities, which have the potential to stabilize combustion without excessive total pressure loss and are therefore used as flameholders in supersonic combustors. In the present study, we perform high-fidelity large-eddy simulations (LES) of a hydrogen jet in a supersonic crossflow (JISCF) of vitiated air, which is located upstream of a wall-mounted squared cavity. The performance of such high-fidelity LES does not only require the use of high-precision numerical schemes and reliable subgrid-scale models relevant to the so-called direct numerical simulation (DNS) limit, it is also strongly dependent on the mesh quality. Therefore, the present study places special emphasis on computational grid assessment through the introduction of a detailed numerical procedure, which aims at analysing mesh reliability. The corresponding procedure combines several verification subsets including (i) the inspection of distributions of the dimensions of the computational cells present at the wall location, (ii) the analysis of normalized velocity profiles and viscosity ratio in boundary layers, and (iii) the check of fields of some mesh quality indexes and associated distributions. For the geometry under consideration, it appears that the level of resolution imposed by a correct description of boundary layers leads to a mesh quality that is close to the one associated to DNS requirements. Combustion stabilization is then studied for two distinct values of the inlet vitiated airstream temperature. Two stabilization modes are recovered from the numerical simulations: cavity-stabilized and jet-wake stabilized regimes.