On coincidence of Alday-Maldacena-regularized $\sigma$-model and Nambu-Goto areas of minimal surfaces

For the $\sigma$-model and Nambu-Goto actions, values of the Alday-Maldacena-regularized actions are calculated on solutions of the equations of motion with constant non-regularized Lagrangian. It turns out that these values coincide up to a factor, independent of boundary conditions.