  The exploration of the phase diagram of Quantum ChromoDynamics (QCD) is
carried out by studying ultrarelativistic heavy-ion collisions. The energy
range covered by the CERN SPS ($\sqrt{s_{\rm \scriptscriptstyle{NN}}} \sim$
6-17 GeV) is ideal for the investigation of the region of the phase diagram
corresponding to finite baryochemical potential ($\mu_{\rm B}$), and has been
little explored up to now. We propose in this document a new experiment, NA60+,
that would address several observables which are fundamental for the
understanding of the phase transition from hadronic matter towards a
Quark-Gluon Plasma (QGP) at SPS energies. In particular, we propose to study,
as a function of the collision energy, the production of thermal dimuons from
the created system, from which one would obtain a caloric curve of the QCD
phase diagram that is sensitive to the order of the phase transition. In
addition, the measurement of a $\rho$-a$_1$ mixing contribution would provide
conclusive insights into the restoration of the chiral symmetry of QCD. In
parallel, studies of heavy quark and quarkonium production would also be
carried out, addressing the measurement of transport properties of the QGP and
the investigation of the onset of the deconfinement transition. The document
also defines an experimental set-up which couples a vertex telescope based on
monolithic active pixel sensors (MAPS) to a muon spectrometer with tracking
(GEM) and triggering (RPC) detectors within a large acceptance toroidal magnet.
Results of physics performance studies for most observables accessible to NA60+
are discussed, showing that the results of the experiment would lead to a
significant advance of our understanding of strong interaction physics. The
document has been submitted as an input to the European Particle Physics
Strategy Update 2018-2020 (http://europeanstrategyupdate.web.cern.ch/).
