Here I assume that conventional superconductors refer to materials that undergo a superconducting transition at low temperature as described by the Bardeen-Cooper-Schrieffer theory (BCS). For those materials the effective attractive electron-electron interaction is mediated by phonons. How accurately can the superconducting transition temperature be computed from ab initio methods and what are the factors that limit the accuracy of those methods?
I am not an expert in superconductivity but I believe that the crucial interaction that dictate the conventional superconductivity is the phonon-electron interaction which in the language of quantum chemistry can be regarded as a non-adiabatic nuclear dynamics which is beyond Bohr-Oppenheimer approximation. And I do not think conventional electronic calculations incorporate this effect.