  We investigate theoretically the superconducting state of the undoped
Fe-based superconductor ThFeAsN. Using input from $ab~initio$ calculations, we
solve the Fermi-surface based, multichannel Eliashberg equations for
Cooper-pair formation mediated by spin and charge fluctuations, and by the
electron-phonon interaction (EPI). Our results reveal that spin fluctuations
alone, when coupling only hole-like with electron-like energy bands, can
account for a critical temperature $T_c$ up to $\sim7.5\,\mathrm{K}$ with an
$s_{\pm}$-wave superconducting gap symmetry, which is a comparatively low $T_c$
with respect to the experimental value $T_c^{\mathrm{exp}}=30\,\mathrm{K}$.
Other combinations of interaction kernels (spin, charge, electron-phonon) lead
to a suppression of $T_c$ due to phase frustration of the superconducting gap.
We qualitatively argue that the missing ingredient to explain the gap magnitude
and $T_c$ in this material is the first-order correction to the EPI vertex. In
the noninteracting state this correction adopts a form supporting the $s_{\pm}$
gap symmetry, in contrast to EPI within Migdal's approximation, i.e., EPI
without vertex correction, and therefore it enhances tendencies arising from
spin fluctuations.
