We present a dielectric study of H-H mesophases (H-H) based on a GMO/tricaprylin/phosphatidylcholinc/water system seeded with the peptide Cyclosporine A (CSA). The study covers a frequency range 0.01 Hz to 1 MHz and a temperature range of 293 to 319 K, with a 3 K temperature step. Three dielectric relaxation processes are observed and discussed. This picture is further elucidated by comparison with a dielectric study of the empty H-H mesophase system, previously published, where the same three processes were involved. A complex picture emerges whereby the CSA is intercalated between the surfactant tails yet protrudes into the interface as well. Whereas the CSA remains hydrophobic, it still influences the relaxation behavior of the GMO head and counterion movement along the interface in a nontrivial manner. The third dipolar species, the tricaprylin molecule, is also influenced by the presence of CSA. A critical temperature T-0 = 307 K is recognized and identified as the dehydration temperature of the surfactant heads. This induces a conformal transition in the GSA, drastically changing its effect on the three dielectric processes evident in the raw data. The implications of this behavior are discussed in detail.