Context. One of the primary goals of Galactic Archaeology is to reconstruct the Milky Way's accretion history. To achieve this, significant efforts have been dedicated to identifying signatures of past accretion events. In particular, the study of the integrals of motion (IoM) space has proven to be highly insightful for uncovering these ancient mergers and understanding their impact on the Galaxy's evolution. Aims. This paper evaluates the effectiveness of a state-of-the-art method for detecting debris from accreted galaxies by testing it on four Milky Way-like galaxies from the Auriga suite of cosmological magnetohydrodynamic simulations. Methods. We employed an innovative method to identify substructures in the IoM space within the local stellar halos of the four simulated galaxies. This approach enabled us to evaluate the method's performance by comparing the properties of the identified clusters with the known populations of accreted galaxies in the simulations. Additionally, we investigated whether incorporating chemical abundances and stellar age information can help to link distinct structures originating from the same accretion event. Results. This method is very effective in detecting debris from accretion events occurring less than 6─7 Gyr ago, but it struggles to detect most of the debris from older accretion. Furthermore, most of the detected structures suffer from significant contamination from in situ stars. Our results also show that the method might also generate artificial detections. Conclusions. Our work shows that the Milky Way's accretion history remains uncertain, while questioning the reality of some of the structures detected in the solar vicinity.