  The mechanism that restores the pseudo-spin symmetry (PSS) are investigated
under the relativistic Hartree-Fock (RHF) approach, by focusing on the
in-medium balance between nuclear attractive and repulsive interactions. It is
illustrated that the modelings of both the equilibrium of nuclear dynamics and
the in-medium effects can be essentially changed by the $\rho$-tensor coupling
that play the role almost fully via the Fock terms, from which the model
discrepancy on the PSS restoration is verified. Specifically, the largely
different density-dependent behaviors of the isoscalar coupling strengths
$g_\sigma$ and $g_\omega$, deduced from the parametrization of the RHF
Lagrangian PKA1, play an essential role in restoring the PSS of the high-$l'$
pseudo-spin doublets around the Fermi levels. Qualitatively, a guidance is
provided for the modelings of both the equilibrium of nuclear dynamics and the
in-medium effects via the PSS restoration.
