The NA60+ experiment at the CERN SPS to study dilepton and heavy quark production at large $\mu_{B}$

The region of high baryonic densities of the QCD phase diagram is the object of several studies focused on the investigation of the order of the phase transition and the search for the critical point. The rare probes, which include electromagnetic observables and heavy quark production, are experimentally challenging to access as they require large integrated luminosities that could be studied with fixed-target experiments. A future experiment, NA60+ at CERN, is being proposed to access this region and perform accurate measurements of the dimuon spectrum up to the charmonium region and study charm and strange hadrons. With its high beam intensity, the CERN SPS can cover the center-of-mass collision energy region from 6 to 17 GeV providing access to rare observables which have been scarcely studied until now. The proposed experiment includes a muon spectrometer based on tracking gas detectors coupled to a vertex spectrometer based on Si detectors. The time slot after the Long Shutdown 3 of the LHC (past 2029) is aimed for the first data-taking, with Pb and proton beams. In this contribution, we review the project and recent R&D effort, including the technical aspects and the studies of the physics performances for the observables.