Context. Recently, the colliding-wind region of the binary stellar system HD 93129A was resolved for the first time using Very Large Baseline Interferometry. This system, one of the most massive known binaries in our Galaxy, presents non-thermal emission in the radio band, which can be used to infer the physical conditions in the system, and make predictions for the high-energy band. Aims. We intend to constrain some of the unknown parameters of HD 93129A through modeling the non-thermal emitter. We also aim to analyse the detectability of this source in hard X-rays and γ-rays. Finally, we want to predict how the non-thermal emission will evolve in the future, when the stars approach periastron. Methods. A broadband radiative model for the wind-collision region (WCR) has been developed taking into account the evolution of the accelerated particles streaming along the shocked region, the emission by different radiative processes, and the attenuation of the emission propagating through the local matter and radiation fields. We reproduce the available radio data, and make predictions of the emission in hard X-rays and γ-rays under different assumptions.Results. From the analysis of the radio emission, we find that the binary HD 93129A is more likely to have a low inclination and a high eccentricity, with the more massive star being currently closer to the observer. The minimum energy of the non-thermal electrons seems to be between ∼20-100 MeV, depending on the intensity of the magnetic field in the WCR. The latter can be in the range ∼20-1500 mG. Conclusions. Our model is able to reproduce the observed radio emission, and predicts that the non-thermal radiation from HD 93129A will increase in the near future. With instruments such as NuSTAR, Fermi, and CTA, it will be possible to constrain the relativistic particle content of the source, and other parameters such as the magnetic field strength in the WCR which, in turn, can be used to obtain upper-limits of the magnetic field on the surface of the very massive stars, thereby inferring whether magnetic field amplification is taking place in the particle acceleration region.