  We investigate the photon-mediated heat flow between two Josephson-coupled
Bardeen-Cooper-Schrieffer (BCS) superconductors. We demonstrate that in
standard low temperature experiments involving temperature-biased
superconducting quantum interference devices (SQUIDs), this radiative
contribution is negligible if compared to the direct galvanic one, but it
largely exceeds the heat exchanged between electrons and the lattice phonons.
The corresponding thermal conductance is found to be several orders of
magnitude smaller, for real experiments setup parameters, than the universal
quantum of thermal conductance, kappa_0(T)=pi k_B^2T/6hbar.
