Information about research on superconductivity and plasmas based on the Darwin Hamiltonian approach can be found in the following papers and links. The main result is that superconductivity is the phase transition caused by the Darwin (-Breit) term in the Hamiltonian, i.e. the attraction between parallel currents. Alternatively, the superconducitvity phase transition may be said to be caused by the energy lowering resulting from the correlation of electron momenta on the Fermi surface. Phonons only play a destructive role and partly determine the transition temperature by destroying the correlation of electron momenta at higher temperatures.

A short history of the idea:

The famous russian physicist J. Frenkel (Yacov Ilich Frenkel, 1894-1952) originally suggested the magnetic explanation of superconductivity in 1933. The same year, 1933, Hans Bethe and H. Fröhlich published a paper claiming that Frenkel's explanation had to be wrong. Their argument is classical and based on a version of the Darwin Hamiltonian (Charles Galton Darwin, 1887-1962) derived by the authors, independently of Darwin (1920). They certainly did not prove that Frenkel was wrong. Instead they proved that the handling of magnetic interaction energy in the solid state is problematic. Later the german physicist Heinrich Welker (1912-1981) again suggested a magnetic explanation of superconductivity (1939). He tried to get around the problems suggested by Bethe and Fröhlich by discussing quantum effects, in particular exchange. Welker also gave an elegant explanation of the Meissner effect (1938). Thus ended the early attempts at a magnetic theory of superconductivity.
As I see it the error in the Bethe-Fröhlich paper is the purely classical treatment. Electrons produce and interact with magnetic fields via the expectation value of their current density. The majority of electrons in a metal occupy states with real wave functions (zero current density) and thus neither produce nor interact with a magnetic field. The remaining few that do are the ones that are part of the superconducting condensate. I therefore think that Fröhlich and BCS are wrong; phonons do not cause superconductivity, they destroy it. Magnetic interaction is the true cause, but no one knows how to calculate accurately and quantum mechanically, the properties of systems of very many particles interacting magnetically. Order of magnitude estimates and simple approximations and idealizations, however, support the Frenkel-Welker-Essén theory.