[en] We study the null-polygonal minimal surfaces in AdS₄, which correspond to the gluon scattering amplitudes/Wilson loops in N=4 super Yang-Mills theory at strong coupling. The area of the minimal surfaces with n cusps is characterized by the thermodynamic Bethe ansatz (TBA) integral equations or the Y-system of the homogeneous sine-Gordon model, which is regarded as the SU(n-4)₄/U(1)^n-5 generalized parafermion theory perturbed by the weight-zero adjoint operators. Based on the relation to the TBA systems of the perturbed W minimal models, we solve the TBA equations by using the conformal perturbation theory, and obtain the analytic expansion of the remainder function around the UV/regular-polygonal limit for n = 6 and 7. We compare the rescaled remainder function for n=6 with the two-loop one, to observe that they are close to each other similarly to the AdS₃ case.