Minkowski content estimates for generic area minimizing hypersurfaces

Let $\Gamma$ be a smooth, closed, oriented, $(n-1)$-dimensional submanifold of $\mathbb{R}^{n+1}$. It was shown by Chodosh-Mantoulidis-Schulze that one can perturb $\Gamma$ to a nearby $\Gamma'$ such that all minimizing currents with boundary $\Gamma'$ are smooth away from a set with Hausdorff dimension less than $n-9$. We prove that the perturbation can be made such that the singular set of the minimizing current with boundary $\Gamma'$ has Minkowski dimension less than $n-9$.