Energy of the magnetic field

I have a doubt regarding a particular problem.
Let’s suppose to have a hollow vertical cylinder whit a vertical current flowing on the surface. Let’s now suppose to compress the cylinder so that the radius changes of $dR$, but there is a battery that keeps the current costant. Now in a certain region of space there is a magnetic field that wasn’t there before. I expect a variation of energy stored in the magnetic field of $dU=frac{1}{2mu_0}B^2dV$. So i must have a work done from the outside. My question is essentialy who does this force? During the compression the charges flowing have a oblique velocity. So the Lorentz force is oblique itself and we have a vertical and a horizontal component. I expect that in order to perform this compression we have to do two things. We must have a horizontal force opposite to the horizontal component of Lorentz force, and the battery must do a vertical force in order to sustain the current against the vertical component of Lorentz force. But this way the sum of the two externel forces makes no work, because it’s perpendicular to the velocity. We have the correct result if we consider just one of the two forces, but why? I’m totally missing something.

Thank you for your help

Sincerely