Engel et al.'s antichain Hausdorff-measure conjecture

Let $[0,1]^n$ be equipped with the coordinatewise partial order, and let an antichain be a subset $A\subset[0,1]^n$ containing no two distinct comparable points. Write $\mathcal{H}^{n-1}(A)$ for its $(n-1)$-dimensional Hausdorff measure. Engel et al.'s conjecture. There exists an antichain $A$ in $[0,1]^n$ such that

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

The bound $\mathcal{H}^{n-1}(A)\leq n$ is known, and it is asymptotically sharp; the conjecture is verified for $n=1$ and $n=2$, but its validity in general remains open.