Almost-everywhere sublevel sets of MA functions belong to \mathcal N

Let $\operatorname{MA}(\mathbb{R}^d)$ denote the class used in the source, and let $\mathcal N$ be the class of compact sets admitting decreasing smooth-domain approximations with uniformly bounded normal-cycle masses. Sublevel-set conjecture. For $f\in\operatorname{MA}(\mathbb{R}^d)$, one has

$$f^{-1}((-\infty,c])\in\mathcal N$$

for almost every $c\in\mathbb{R}$. This is presented as a possible extension of the approach for subanalytic sets; the statement remains open.