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Context. Rubidium is one of the few elements produced by the neutron capture s - and r -processes in almost equal proportions. Recently, a Rb deficiency ([Rb/Fe] < 0.0), amounting to a factor of about two with respect to the Sun, has been found in M dwarfs of near-solar metallicity. This stands in contrast to the close-to-solar [Sr, Zr/Fe] ratios derived in the same stars. This deficiency is difficult to understand from the point of view of observations and of nucleosynthesis. Aims. To test the reliability of this Rb deficiency, we study the Rb and Zr abundances in a sample of KM-type giant stars across a similar metallicity range, extracted from the AMBRE Project. Methods. We used high-resolution and high signal-to-noise spectra to derive Rb and Zr abundances in a sample of 54 bright giant stars with metallicities in the range of −0.6 ≲ [Fe/H] ≲ +0.4 dex, via spectral synthesis in both local and non-local thermodynamic equilibrium (LTE and NLTE, respectively). We also studied the impact of the Zeeman broadening in the profile of the Rb I at λ 7800 Å line. Results. The LTE analysis also results in a Rb deficiency in giant stars, however, it is considerablymore »