We examine the reliability of the cumulative thermal conductivity as a function of the free path, F(Λ), in the context of reconstruction of phonon transport properties from thermal spectroscopy experiments. We specifically show that a given F(Λ)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(Λ). Since different relaxation-time distributions will, in general, lead to different thermal responses, F(Λ) 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.