Improved generic regularity of codimension-1 minimizing integral currents

Let $\mathrm{\Gamma }$ be a smooth, closed, oriented, $(n-1)$-dimensional submanifold of ${\mathbb{R}}^{n+1}$. We show that there exist arbitrarily small perturbations ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ of $\mathrm{\Gamma }$ with the property that minimizing integral $n$-currents with boundary ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ are smooth away from a set of Hausdorff dimension $\le n-9-{\epsilon }_n$, where ${\epsilon }_n\in (0,1]$ is a dimensional constant. This improves on our previous result (where we proved generic smoothness of minimizers in $9$ and $10$ ambient dimensions). The key ingredients developed here are a new method to estimate the full singular set of the foliation by minimizers and a proof of superlinear decay of closeness (near singular points) that holds even across non-conical scales.