The primary alpha and the secondary Johari-Goldstein (JG) beta relaxations of supercooled glass-forming neat epoxy resin and 2-picoline in mixture with tristyrene are monitored by broadband dielectric relaxation spectroscopy at ambient pressure and elevated pressures. For different combinations of pressure and temperature that maintain the alpha-relaxation time constant, the frequency dispersion of the alpha relaxation is unchanged, as previously found in other glass-formers, but remarkably the JG beta-relaxation time remains constant. This is more clear evidence of a strong connection between the alpha- and JG beta-relaxation times, a fact that should be taken into account in the construction of a viable theory of glass transition.