I don't hope too much to get the answer to the electric current density Ehrenfest paradox. But, let me ask a simple part of it. Why discussions of relativistic origin of the magnetic field never discuss the simplest case: a neutral wire with current and test charge at rest?

They say that moving charge relativistically gets extra charge density and, therefore, wire becomes electrically charged (negatively, because electrons are moving). It will attract or repel electrical test charges. This is what should happen if only one polarity charges are moving in the wire. Right? To stay neutral, the relativistic charge density should be balanced by the current of positive charge! Positive ions must move to compensate the motion of free electrons!

Yet, the relativistic magnetism discussions avoid discussing this curios fact. They jump immediately to the case when there is electric balance for a test charge in motion (e.g. here, here and here). The second also makes a confession that

In the frame in which the wire is at rest, the positive and negative charge densities exactly balance, otherwise there will be extra electrostatic fields

It considers the lab frame as a case where positive and negative charges move in opposite directions so that net current is not zero. But, it forgets to mention that in the normal lab, the positive charges create a solid structure of the frame and, therefore, cannot move. And that is the contradiction, I am paying attention on.

I have revised the Feynman lectures. He considers test charge and electrons at rest, telling that wire is charged positively. This seems to confirm my idea that current of electrons charges the neutral wire negatively. But, he adds immediately that this does not happen because if it would be true then, temperature would charge the bodies since electrons are much lighter and gain more speed when body heated. But, we observe no charge variations as temperature varies. This is another contradiction that I ask to resolve along with the issue why nobody can say explicitly whether current makes wire charged or not.