Coaxial cable with infinite return conductor

If a coaxial cable has a coaxial return conductor with infinite outer radius, will the return conductor experience a voltage build-up due to current flowing through it, or will it stay on ground potential? Here I’m taking infinite to be the zero potential.

I would appreciate if both the d.c. and a.c. case are discussed, if the answer is dependent on type of excitation.

Edit: I think the specific problem where my intuition lets me down is whether there is a scalar potential build-up axially along the return conductor, such that the electric field in the return conductor is given by

$$
mathbf{E} = -nabla varphi – frac{partial mathbf{A}}{partial t}
$$

or, if the electric field is only due to the changing magnetic field, i.e. $mathbf{E} = – frac{partial mathbf{A}}{partial t}$.

My reason for thinking it’s the latter is that since the return conductor certainly is a good conductor, then $varphi$ must be constant and equal to zero, or else currents would flow radially to cancel the charge/potential build-up. Is this correct?