The even-modulus regularity conjecture for area-minimizing currents

Let $T$ be as in Theorem. For odd $p$, Theorem states that $\operatorname{Sing}(T)$ is $(m-1)$-rectifiable and has locally finite $(m-1)$-dimensional Hausdorff measure away from $\operatorname{spt}^{p}(\partial T)$.

Even-modulus regularity conjecture. The conclusions of Theorem hold for $p$ even as well; that is, when $p$ is even, $\operatorname{Sing}(T)$ is $(m-1)$-rectifiable and, for every compact $K$ with $K\cap\operatorname{spt}^{p}(\partial T)=\varnothing$,

$$\mathcal{H}^{m-1}(\operatorname{Sing}(T)\cap K)<\infty.$$

For odd $p$ these conclusions are proved in the paper, whereas the even-$p$ case is presented as an open extension and would give a uniform rectifiability and measure theory for singular sets of area-minimizing currents modulo every $p$.