  The two recent density-dependent versions of the finite-range M3Y interaction
(CDM3Y$n$ and M3Y-P$n$) have been probed against the bulk properties of
asymmetric nuclear matter (NM) in the nonrelativistic Hartree Fock (HF)
formalism. The same HF study has also been done with the famous Skyrme (SLy4)
and Gogny (D1S and D1N) interactions which were well tested in the nuclear
structure calculations. Our HF results are compared with those given by other
many-body calculations like the Dirac-Brueckner Hartree-Fock approach or
ab-initio variational calculation using free nucleon-nucleon interaction, and
by both the nonrelativistic and relativistic mean-field studies using different
model parameters. Although the two considered density-dependent versions of the
M3Y interaction were proven to be quite realistic in the nuclear structure or
reaction studies, they give two distinct behaviors of the NM symmetry energy at
high densities, like the Asy-soft and Asy-stiff scenarios found earlier with
other mean-field interactions. As a consequence, we obtain two different
behaviors of the proton fraction in the $\beta$-equilibrium which in turn can
imply two drastically different mechanisms for the neutron star cooling. While
some preference of the Asy-stiff scenario was found based on predictions of the
latest microscopic many-body calculations or empirical NM pressure and isospin
diffusion data deduced from heavy-ion collisions, a consistent mean-field
description of nuclear structure database is more often given by some Asy-soft
type interaction like the Gogny or M3Y-P$n$ ones. Such a dilemma poses an
interesting challenge to the modern mean-field approaches.
