We present results from small-angle X-ray scattering and turbidity measurements on the effect of high hydrostatic pressure on the phase behavior of dense lysozyme solutions in the liquid-liquid phase separation region, and characterize the underlying intermolecular protein-protein interactions as a function of temperature and pressure under charge-screening conditions (0.5 M NaCl). A reentrant liquid-liquid phase separation region is observed at elevated pressures, which may originate in the pressure dependence of the solvent-mediated protein-protein interaction. A temperature-pressure-concentration phase diagram was constructed for highly concentrated lysozyme solutions over a wide range of temperatures, pressures and protein concentrations including the critical region of the liquid-liquid miscibility gap.