Molecular dynamics (MD) was employed by means of a specific simulation protocol to investigate the equilibrium structure at 25 °C of the hexagonal inverted (HII) mesophase composed from water, 1-monoolein (GMO), and tricaprylin, with or without entrapped lysozyme. Based on robust and fast MD simulations, the study provides a comprehensive analysis and visualization of the local structure of HII mesophase containing admixtures. The most important physical insight is the possibility to observe the strong self-recovery capacity of the GMO layer, which allows the HII mesophase tubes to reorganize and host lysozyme molecules with a size bigger than the diameter of the water channel. This is a direct message to the experimenters that the HII mesophase has the potential to host molecules larger than the diameter of the water channel. Collective character of the interlipid interactions is outlined, which is not affected by the presence of the cargo and may be the reason for the efficient GMO reorganization. Another important result is the possible explanation of the role of triacylglycerols on the low-temperature stabilization of the HII mesophase. The analysis shows that despite the low amount of tricaprylin, its molecules prevent the extreme inclination of the lipid tails and thus optimize the alignment capacity of the lipid tails layer. The study also reveals that the packing frustration does not depend on the temperature and the presence of admixtures. Hence, it might be numerically defined as a universal invariant parameter of a stable HII mesophase composed of a certain lipid.