  Within the newly updated version of the Ultra-relativistic quantum molecular
dynamics (UrQMD) model in which the Skyrme potential energy-density functional
is introduced, the yield ratio between $^3$H and $^3$He clusters emitted from
central $^{40}$Ca+$^{40}$Ca, $^{96}$Zr+$^{96}$Zr, $^{96}$Ru+$^{96}$Ru, and
$^{197}$Au+$^{197}$Au collisions in the beam energy range from 0.12 to 1
GeV$/$nucleon is studied. The recent FOPI data for the $^3$H$/$$^3$He ratio are
compared with UrQMD calculations using 13 Skyrme interactions (all exhibiting
similar values of iso-scalar incompressibility but very different density
dependences of the symmetry energy). It is found that the $^3$H$/$$^3$He ratio
is sensitive to the nuclear symmetry energy at sub-saturation densities. Model
calculations with moderately soft to linear symmetry energies are in agreement
with the experimental FOPI data. This result is in line with both, the recent
constraints on the low-density symmetry energy available in the literature and
our previous results for the high-density symmetry energy obtained with the
elliptic flow of free nucleons and hydrogen isotopes as a sensitive probe.
