  We show that for an area minimizing $m$-dimensional integral current $T$ of
codimension at least 2 inside a sufficiently regular Riemannian manifold, the
upper Minkowski dimension of the interior singular set is at most $m-2$. This
provides a strengthening of the existing $(m-2)$-dimensional Hausdorff
dimension bound due to Almgren and De Lellis & Spadaro. As a by-product of the
proof, we establish an improvement on the persistence of singularities along
the sequence of center manifolds taken to approximate $T$ along blow-up scales.
