  4FGL J1405.1-6119 is a high-mass $\gamma$-ray emitting binary that has been
studied at several wavelengths. The nature of this type of binary is still
under debate, with three possible scenarios usually invoked to explain the
origin of the $\gamma$-ray emission: collisions between the winds of a rapidly
rotating neutron star and its companion, collisions between the winds of two
massive stars, and non-thermal emission from the jet of a microquasar. We
analyze two pairs of simultaneous NuSTAR and XMM-Newton observations to
investigate the origin of the radio, X-ray, and $\gamma$-ray emissions. We
extracted light curves between 0.5-78 keV from two different epochs, named
Epoch 1 and Epoch 2, respectively. We propose a scenario to explain the
observations involving a parabolic, mildly relativistic, lepto-hadronic jet.
This jet has a compact acceleration region that injects a hard spectrum of
relativistic particles. The dominant non-thermal emission processes include
synchrotron radiation of electrons, inverse Compton scattering of photons from
the stellar radiation field, and the decay of neutral pions resulting from
inelastic proton-proton collisions within the bulk matter of the jet. These
estimates are in accordance with the values of a super-Eddington lepto-hadronic
jet scenario. The compact object could be either a black hole or a neutron star
with a low magnetic field. Most of the X-ray emission from the disk could be
absorbed by the dense wind that is ejected from the same disk. We conclude that
it is possible that the binary 4FGL J1405.1-6119 could be a supercritical
microquasar like SS433.
