The crystal-field ground state wave function of CeCu$_2$Si$_2$ has been investigated with linear polarized $M$-edge x-ray absorption spectroscopy from 250mK to 250K, thus covering the superconducting ($T_{text{c}}$=0.6K), the Kondo ($T_{text{K}}$$approx$20K) as well as the Curie-Weiss regime. The comparison with full-multiplet calculations shows that the temperature dependence of the experimental linear dichroism is well explained with a $Gamma_7^{(1)}$ crystal-field ground-state and the thermal population of excited states at around 30meV. The crystal-field scheme does not change throughout the entire temperature range thus making the scenario of orbital switching unlikely. Spectroscopic evidence for the presence of the Ce 4$f^0$ configuration in the ground state is consistent with the possibility for a multi-orbital character of the ground state. We estimate from the Kondo temperature and crystal-field splitting energies that several percents of the higher lying $Gamma_6$ state and $Gamma_7^{(2)}$ crystal-field states are mixed into the primarily $Gamma_7^{(1)}$ ground state. This estimate is also supported by re-normalized band-structure calculations that uses the experimentally determined crystal-field scheme.
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