In the spectral density of Cr(001) surfaces, a sharp resonance close to the Fermi level is observed in both experiment and theory. For the physical origin of this peak, two mechanisms were proposed: a single-particle $d_{z2}$ surface state renormalized by electron-phonon coupling and an orbital Kondo effect due to the degenerate $d_{xz}/d_{yz}$ states. Despite several experimental and theoretical investigations, the origin is still under debate. In this work, we address this problem by two different approaches of the dynamical mean-field theory: first, by the spin-polarized $T$-matrix fluctuation exchange approximation suitable for weakly and moderately correlated systems; second, by the noncrossing approximation derived in the limit of weak hybridization (i.e., for strongly correlated systems) capturing Kondo-type processes. By using recent continuous-time quantum Monte Carlo calculations as a benchmark, we find that the high-energy features, everything except the resonance, of the spectrum are captured within the spin-polarized $T$-matrix fluctuation exchange approximation. More precisely, the particle-particle processes provide the main contribution. For the noncrossing approximation, it appears that spin-polarized calculations suffer from spurious behavior at the Fermi level. Then, we turned to non-spin-polarized calculations to avoid this unphysical behavior. By employing two plausible starting hybridization functions, it is observed that the characteristics of the resonance are crucially dependent on the starting point. It appears that only one of these starting hybridizations could result in an orbital Kondo resonance in the presence of a strong magnetic field like in the Cr(001) surface. It is for a future investigation to first resolve the unphysical behavior within the spin-polarized noncrossing approximation and then check for an orbital Kondo resonance.