Thermal transport properties of amorphous materials at low temperatures are governed by the interaction between phonons and localized excitations referred to as tunneling two-level systems (TLSs). The temperature variation of the thermal conductivity of these amorphous materials is considered as universal and is characterized by a quadratic power law. This is well described by the phenomenological TLS model even though its microscopic explanation is still elusive. Here, by scaling down to the nanometer-scale amorphous systems much below the bulk phonon-TLS mean free path, we probe the robustness of that model in restricted geometry systems. Using very sensitive thermal conductance measurements, we demonstrate that the temperature dependence of the thermal conductance of silicon nitride nanostructures remains mostly quadratic independently of the nanowire section. It does not follow the cubic power law in temperature as expected in a Casimir-Ziman regime of boundary-limited thermal transport. This shows a thermal transport counterintuitively dominated by phonon-TLS interactions and not by phonon boundary scattering in the nanowires. This could be ascribed to an unexpected high density of TLSs on the surfaces which still dominates the phonon diffusion processes at low temperatures and explains why the universal quadratic temperature dependence of thermal conductance still holds for amorphous nanowires.