  The energy-influx/entropy-influx relation in the Green-Naghdi Type III theory
of heat conduction is examined within a thermodynamical framework \`a la
Mueller-Liu, where that relation is not specified a priori irrespectively of
the constitutive class under attention. It is shown that the classical
assumption, i.e., that the entropy influx and the energy influx are
proportional via the absolute temperature, holds true if heat conduction is, in
a sense that is made precise, isotropic. In addition, it is proven that the
standard assumption does not hold in case of transversely isotropic conduction.
