The flat-point Hausdorff-dimension conjecture for area-minimizing currents mod p

Let $T$ be as in Theorem. Denote by $\operatorname{Sing}_{f}(T)$ the subset of interior flat singular points, namely those points $q\in \operatorname{Sing}(T)$ at which there is at least one flat tangent cone.

Flat-point conjecture.

$$\mathcal{H}^{m-2+\alpha}(\operatorname{Sing}_{f}(T))=0$$

for every $\alpha>0$.

This conjecture concerns the size of the singular set at points admitting a flat tangent cone and would extend the known codimension-two Hausdorff-dimension estimate in the classical mod-$2$ case. It is proved in the paper for every odd $p$, but remains open for even $p$ in the generality stated.