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  1. 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 »lower than that obtained in M dwarfs. However, once NLTE corrections are performed, the [Rb/Fe] ratios are very close to solar (average −0.01 ± 0.09 dex) in the full metallicity range studied here. This stands in contrast to the value found for M dwarfs. The [Zr/Fe] ratios derived are in excellent agreement with those obtained in previous studies in FGK dwarf stars with a similar metallicity. We investigate the effect of gravitational settling and magnetic activity as possible causes of the Rb deficiency found in M dwarfs. Although the former phenomenon has a negligible impact on the surface Rb abundance, the presence of an average magnetic field with an intensity that is typical of that observed in M dwarfs may result in systematic Rb abundance underestimations if the Zeeman broadening is not considered in the spectral synthesis. This may explain the Rb deficiency in M dwarfs, but not fully. On the other hand, the new [Rb/Fe] and [Rb/Zr] versus [Fe/H] relationships can be explained when the Rb production by rotating massive stars and low-to-intermediate mass stars (these latter also producing Zr) are considered, without the need to deviate from the standard s -process nucleosynthesis in asymptotic giant branch stars, as suggested previously.« less