In this work, we extend, review and jointly discuss earlier experiments conducted by us in hyperaged geological glasses, either in Dominican amber (20 million years old) or in Spanish amber from El Soplao (110 million years old). After characterization of their thermodynamic and elastic properties (using Differential Scanning Calorimetry around the glass-transition temperature, and measuring mass density and sound velocity), their specific heat was measured at low and very low temperatures. By directly comparing pristine amber samples (i.e. highly stabilized polymer glasses after ageing for millions of years) to the same samples after being totally or partially rejuvenated, we have found that the two most prominent universal "anomalous" low-temperature properties of glasses, namely the tunnelling two-level systems and the so-called "boson peak", persist essentially unchanged in both types of hyperaged geological glasses. Therefore, non-Debye low-energy excitations of glasses appear to be robust, intrinsic properties of non-crystalline solids which do not vanish by accessing to very deep states in the potential energy landscape.