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Relaxing the conventional assumption of a minimal coupling between the dark matter (DM) and dark energy (DE) fields introduces significant changes in the predicted evolution of the Universe. Therefore, testing such a possibility constitutes an essential task not only for cosmology but also for fundamental physics. In a previous communication [Phys. Rev. D99, 043521, 2019], we proposed a new null test for the $Λ$CDM model based on the time dependence of the ratio between the DM and DE energy densities which is also able to detect potential signatures of interaction between the dark components. In this work, we extend that analysis avoiding the $ Λ$CDM assumption and reconstruct the interaction in the dark sector in a fully model-independent way using data from type Ia supernovae, cosmic chronometers and baryonic acoustic oscillations. According to our analysis, the $Λ$CDM model is consistent with our model-independent approach at least at $3σ$ CL over the entire range of redshift studied. On the other hand, our analysis shows that the current background data do not allow us to rule out the existence of an interaction in the dark sector. Finally, we present a forecast for next-generation LSS surveys. In particular, we show that Euclid and SKA will be able to distinguish interacting models with about 4\% of precision at $z\approx 1$.