  Motivated by the recent report of room-temperature superconductivity at
near-ambient pressure in N-doped lutetium hydride by Dasenbrock et al. [Nature
615, 244 (2023)], we performed a comprehensive, detailed study of the phase
diagram of the Lu-N-H system, looking for superconducting phases. We combined
ab initio crystal structure prediction with ephemeral data-derived interatomic
potentials to sample over 200,000 different structures. Out of the more than
150 structures predicted to be metastable within $\sim$ 50 meV from the convex
hull we identify 52 viable candidates for conventional superconductivity, for
which we computed their superconducting properties from Density Functional
Perturbation Theory. Although for some of these structures we do predict a
finite superconducting $T_{\text{c}}$, none is even remotely compatible with
room-temperature superconductivity as reported by Dasenbrock et al. Our work
joins the broader community effort that has followed the report of near-ambient
superconductivity, confirming beyond reasonable doubt that no conventional
mechanism can explain the reported $T_{\text{c}}$ in Lu-N-H.

Superconductivity;Materials Science