**Abstract** : 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.