  In the case of Li-intercalated hexagonal boron nitride bilayer (Li-hBN), the
vertex corrections of electron-phonon interaction cannot be omitted. This is
evidenced by the very high value of the ratio
$\lambda\omega_{D}/\varepsilon_{F}\sim 0.46$, where $\lambda$ is the
electron-phonon coupling constant, $\omega_{D}$ is the Debye frequency, and the
symbol $\varepsilon_{F}$ represents the Fermi energy. Due to the nonadiabatic
effects, the phonon-induced superconducting state in Li-hBN is characterized by
the much lower value of critical temperature ($T^{\rm LOVC}_{C}\in\{ 19.1,
15.5, 11.8\}$ K, for $\mu^{\star}\in \{0.1, 0.14, 0.2\}$), than would result
from calculations not taking this effect into account: $T^{\rm ME}_{C}\in\{
31.9, 26.9, 21\}$ K. From the technological point of view, the low value of
$T_{C}$ limits the possible applications of Li-hBN superconducting properties.
The calculations were carried out under the classic Migdal-Eliashberg formalism
(ME) and the Eliashberg theory with the lowest-order vertex corrections (LOVC).
