Einstein famously explained the anomalous precession of Mercury by showing that in general relativity elliptic orbits precess even in the 2-body problem. But apparently in the early days of quantum mechanics Sommerfeld refined Bohr's model of the hydrogen atom with circular orbits by not just introducing elliptic orbits, but also assuming that they precess based on special relativity only. "Sommerfeld postulated that only certain eccentricities of ellipse are possible trajectories for the electron... But Sommerfeld does not stop here. His next move is to apply the laws of special relativity to the different possible electron trajectories. In essence, Sommerfeld makes use of the equation for a precessing elliptical orbit, but introduces relativity by making a change to the equation for angular momentum..." (p.16).
If I understand this correctly he solves the 2-body problem in special relativity kinematics but under the classical inverse square law. Is it enough to make ellipses precess? Does this effect explain the anomalous precession of Mercury even without general relativity?