  We examine the reliability of the cumulative thermal conductivity as a
function of the free path, $F(\Lambda)$, in the context of reconstruction of
phonon transport properties from thermal spectroscopy experiments. We
specifically show that a given $F(\Lambda)$ does not correspond to a unique
relaxation-times function, in the sense that more than one distribution of
relaxation-times can result in the same $F(\Lambda)$. Since different
relaxation-time distributions will, in general, lead to different thermal
responses, $F(\Lambda)$ does not uniquely predict the material thermal response
in all transport regimes. This implies that in the context of thermal transport
at the nanoscale within the Boltzmann relaxation-time approximation framework,
the fundamental property that should be reconstructed from the thermal
spectroscopy experiments is the frequency-dependent relaxation-times function
(provided group velocities are known), since it explicitly appears in the
governing equation as the input material property. Extensive global
optimization studies show that a previously proposed formulation for
reconstruction based on the frequency-dependent relaxation-times function
[Physical Review B 94, 155439 (2016)] provides numerically unique solutions.
