  We use the fundamental nonequilibrium steady-state fluctuation symmetry and
derive a condition on the validity of the thermodynamic uncertainty relation
(TUR) in thermal transport problems, classical or quantum alike. We test this
condition and study the breakdown of the TUR in different thermal transport
junctions of bosonic and electronic degrees of freedom. First, we prove that
the TUR is valid in harmonic oscillator junctions. In contrast, in the
nonequilibrium spin-boson model, which realizes many-body effects, it is
satisfied in the Markovian limit, but violations arise as we tune (reduce) the
cutoff frequency of the thermal baths, thus observing non-Markovian dynamics.
Finally, we consider heat transport by noninteracting electrons in a
tight-binding chain model. Here we show that the TUR is feasibly violated by
tuning e.g. the hybridization energy of the chain to the metal leads. These
results manifest that the validity of the TUR relies on the statistics of the
participating carriers, their interaction, and the nature of their couplings to
the macroscopic contacts (metal electrodes, phonon baths).
