Having read that question: Magnetic force as a relativistic effect?
And quoting from the answer: https://physics.stackexchange.com/a/143901/7743
If you want to analyze things in the electrons' rest frame in wire A (whose electrons move at less than half the speed, relative to the wire, as the electrons in wire B), you have to take into account not just the electric force on the electrons in the wire A, which will be repulsive as you say, but also the electric force on the positive charges (the ions that have lost electrons) in wire A, along with the magnetic force on the positive charges in wire A, which are moving in this frame. When I did this in a numerical example below, I did find that the net force on wire A was attractive despite the fact that the electric force on the electrons was repulsive.
So basically the Lorentz force from the viewpoint of the electrons outside the wire can be explained by the difference of apparent charge densities in the wire due to the length contraction of charges moving in different speeds in the wire.
Now what if we remove the positive charges from the picture and deal with a pure electron beam? Could there be an attraction or will it always repel?