Hausdorff-measure lower-bound conjecture for the truncated Laplacian

Let $E \subset \mathbb{R}^n$ be a compact subset of diameter $<R$, fix $k \in \{1,\ldots,n-1\}$ and let $h \in \mathbb{R}^+$ satisfy $hR<k$. Here, $\mu(\mathscr{F}_k^-,E)$ denotes the relevant principal eigenvalue of the truncated Laplacian on $E$, and $\mathscr{H}^k(E)$ is the $k$-dimensional Hausdorff measure of $E$.

Hausdorff-measure lower-bound conjecture. There exists a constant $C>0$, depending on $n$, $k$, $hR$, and an upper bound for $\mathscr{H}^k(E)$, such that

$$\mu(\mathscr{F}_k^-,E) \ge C\mathscr{H}^k(E)^{-\frac{2}{k}}.$$

In particular, if $\mathscr{H}^k(E)=0$, then $\mu(\mathscr{F}_k^-,E)=+\infty$.

This would extend the proved $k=1$ Hausdorff-measure estimate to higher dimensions; the theorem in the paper obtains a nearly sharp bound for $k=2$, with a logarithmic loss.