  Nickelate superconductors are outstanding materials with intriguing analogies
with the cuprates. These analogies suggest that their superconducting mechanism
may be unconventional, although this fundamental question is currently under
debate. Here, we scrutinize the role played by electronic correlations in
enhancing the electron-phonon coupling in the infinite-layer nickelates and the
extent to which this may promote superconductivity. Specifically, we use $ab$
$initio$ many-body perturbation theory to perform state-of-the-art $GW$ and
Eliashberg-theory calculations. We find that the electron-phonon coupling is
effectively enhanced compared to density-functional-theory calculations. This
enhancement may lead to low-$T_c$ superconductivity in the parent compounds
already. However, it remains marginal in the sense that it cannot explain the
record $T_c$s obtained with doping. This circumstance implies that conventional
superconductivity is preempted by another pairing mechanism in the
infinite-layer nickelates.
