  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.
