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 Migdals approximation, i.e., EPI without vertex correction, and therefore it enhances tendencies arising from spin fluctuations.
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