Minkowski content estimates for generic area minimizing hypersurfaces

Let $\mathrm{\Gamma }$ 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 $\mathrm{\Gamma }$ to a nearby ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ such that all minimizing currents with boundary ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ 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 ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ has Minkowski dimension less than $n-9$.