  We present an equilibration-based a posteriori error estimator for
N\'ed\'elec element discretizations of the magnetostatic problem. The estimator
is obtained by adding a gradient correction to the estimator for N\'ed\'elec
elements of arbitrary degree presented in [J. Gedicke, S. Geevers, and I.
Perugia. An equilibrated a posteriori error estimator for arbitrary-order
N\'ed\'elec elements for magnetostatic problems. Journal of Scientific
Computing, 83:1-23, 2020]. This new estimator is proven to be reliable, with
reliability constant 1, and efficient, with an efficiency constant that is
independent of the polynomial degree of the approximation. These properties are
demonstrated in a series of numerical experiments on three-dimensional test
problems.
