NA60+ is a new proposed experiment designed to study the phase diagram of the strongly interacting matter at high baryochemical potentials, ranging from 200 to 550 MeV. It is focused on precision studies of thermal dimuons, heavy quarks, strangeness and hypernuclei production in Pb-Pb collisions.
These processes will be studied as a function of the collision energy with high-intensity Pb beams provided by the CERN SPS, performing an energy scan from $\sqrt{s_{\rm{NN}}}=6$ GeV (corresponding to $E_{\rm lab}\sim$20 AGeV) or even lower, if provided, up to top SPS energy ($\sqrt{s_{\rm{NN}}}=17.3$ GeV, $E_{\rm lab}=$158 AGeV).

The experimental apparatus is composed of a vertex telescope located close to the target and a muon spectrometer located downstream of a hadron absorber. The vertex telescope will consist of several planes of ultra-thin, large area Monolithic Active Pixel sensors (MAPS) embedded in a dipole magnetic field. The muon spectrometer will utilize large area gaseous detectors for muon tracking and a toroidal magnet based on a new light-weight and general-purpose concept.

An ambitious physics program is foreseen, which includes the study of the order of the phase transition at large baryochemical potential through the measurement of a caloric curve and the search for chiral symmetry restoration effects through the $\rho -a_1$ chiral mixing.

This paper will focus on the physics motivations for these measurements based on thermal dimuons. A description of the experimental set-up will be given together with the results of the physics performance studies. Finally, the present status of the project and the outlook will be briefly discussed.