Characterization of the maximal igregularity class by \mathcal N

Let $\mathcal N$ be the class of compact sets $A\subset\mathbb{R}^d$ for which there exists a monotone sequence of compact smooth domains

$$M_1\supset M_2\supset\cdots,\qquad \bigcap_n M_n=A,$$

with the masses of the normal cycles $N_{M_n}$ bounded by a fixed constant. Let an igregularity class be a class of compact subsets satisfying the normal-cycle, diffeomorphism-stability, and almost-everywhere intersection axioms described above. Maximal-class characterization conjecture. The class $\mathcal N$ is the unique maximal igregularity class of subsets of $\mathbb{R}^d$. It is unclear whether WDC sets belong to $\mathcal N$, and the conjecture remains open.