It is shown that above-threshold-ionization peaks disappear when the kinetic energy associated with the nondipole radiation-pressure-induced photoelectron momentum in the laser propagation direction becomes comparable to the photon energy, and how peaks can be made to reappear if knowledge of the length and direction of the photoelectron momentum is at hand and an emission-direction-dependent momentum shift is accounted for. In this sense appropriate analysis of the nondipole effects restores the energy-conserving signature of the above-threshold-ionization peaks. The reported findings should be observable with intense midinfrared laser pulses.