Ah, sorry, forgot to describe such test case. Actually in this
forum thread, there is demos of the method described, and on the screenshot there is FEM brick, attached to world via chain of ball-socket joints, lying on the floor (contact joints with friction - here is where limits introduced). Actually, method is surprisingly good even with quite tight box constraints (e.g. low friction) - can't really share the numbers now, but if you want, I'll try to do some measurements on the weekend.
Actually, floor which produces contacts with friction is switchable in FEM stress demo too (long stiff brick), and it touches the floor when bending, also not really affecting convergence vitally.
One more thing I wanted to add in terms of test cases. We also tested arbitrary FEM models (up to 4k tetras) - they were main source of ill-conditioned systems, but having double Jacobi preconditioner, solver is converging quite fast there too. Models were falling onto floor with friction too, btw