  The superconducting properties of two recently synthesized ternary silicides
with unit formula $X_4$CoSi ($X=$ Nb, Ta) are investigated through \textit{ab
initio} calculations combined with Eliashberg theory. Interestingly, their
crystal structure comprises interlocking honeycomb networks of Nb/Ta atoms.
Nb$_4$CoSi is found to harbor better conditions for phonon-mediated
superconductivity, as it possesses a higher density of states at the Fermi
level, fostering stronger electron-phonon coupling. The superconducting
critical temperatures ($T_c$) follow the same trend, with Nb$_4$CoSi having a
twice higher value than Ta$_4$CoSi. Furthermore, the calculated $T_c$ values
(6.0 K vs. 3.2 K) agree excellently with the experimentally obtained ones,
establishing superconductivity in this new materials class as mediated by the
electron-phonon coupling. Furthermore, our calculations show that the
superconducting properties of these compounds do not simply correlate with the
parameters of their honeycomb networks, contrary to proposals raised in the
literature. Rather, their complete fermiology and phonon spectrum should be
taken into account in order to explain their respective superconducting
properties.
