Sharpness of the antichain Hausdorff measure bound

Let $n\geq 3$, and let an $n$-cube antichain be a subset of $[0,1]^n$ containing no distinct points $\mathbf{x},\mathbf{y}$ with $x_i\leq y_i$ for every $i$. Write $\mathcal{H}^{n-1}$ for $(n-1)$-dimensional Hausdorff measure, and let

$$\sigma_n=\frac{2^{n}\Gamma(n/2+1)}{\pi^{n/2}}.$$

Sharpness conjecture. For every $n\geq 3$, there exists an $n$-cube antichain $S$ such that

$$\mathcal{H}^{n-1}(S)=n\sigma_{n-1}.$$

The smooth-surface result shows that the upper bound $n\sigma_{n-1}$ is asymptotically sharp. This conjecture asks whether the bound is attained exactly in every dimension $n\geq 3$, and remains open.