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Context. The dominant astrophysical production site of the r-process elements has not yet been unambiguously identified. The suggested main r-process sites are core-collapse supernovae and merging neutron stars. Aims. We explore the problem of the production site of Eu. We also use the information present in the observed spread in the Eu abundances in the early Galaxy, and not only its average trend. Moreover, we extend our investigations to other heavy elements (Ba, Sr, Rb, Zr) to provide additional constraints on our results. Methods. We adopt a stochastic chemical evolution model that takes inhomogeneous mixing into account. The adopted yields of Eu from merging neutron stars and from core-collapse supernovae are those that are able to explain the average [Eu/Fe]–[Fe/H] trend observed for solar neighbourhood stars, the solar abundance of Eu, and the present-day abundance gradient of Eu along the Galactic disc in the framework of a well-tested homogeneous model for the chemical evolution of the Milky Way. Rb, Sr, Zr, and Ba are produced by both the s- and r-processes. The r-process yields were obtained by scaling the Eu yields described above according to the abundance ratios observed in r-process rich stars. The s-process contribution by spinstars is the same as in our previous papers. Results. Neutron star binaries that merge in less than 10 Myr or neutron star mergers combined with a source of r-process generated by massive stars can explain the spread of [Eu/Fe] in the Galactic halo. The combination of r-process production by neutron star mergers and s-process production by spinstars is able to reproduce the available observational data for Sr, Zr, and Ba. We also show the first predictions for Rb in the Galactic halo. Conclusions. We confirm previous results that either neutron star mergers on a very short timescale or both neutron star mergers and at least a fraction of Type II supernovae have contributed to the synthesis of Eu in the Galaxy. The r-process production of Sr, Zr, and Ba by neutron star mergers – complemented by an s-process production by spinstars – provide results that are compatible with our previous findings based on other r-process sites. We critically discuss the weak and strong points of both neutron star merging and supernova scenarios for producing Eu and eventually suggest that the best solution is probably a mixed one in which both sources produce Eu. In fact, this scenario reproduces the scatter observed in all the studied elements better.