Magnetostatic edge-element formulation leads to a singular linear system of equations. Although a reduced system of equations can be derived by imposing the zero-tree gauge condition, the gauging causes undesirable deterioration in the convergence property of conventional iterative solvers. In this paper, we develop a new preconditioning technique that overcomes the slow convergence caused by zero-tree gauging, by utilizing the folded preconditioning proposed in our previous paper. Moreover, we present a novel theorem that clarifies the coincidence of the spectra of the preconditioned matrices when folded preconditioning is used.