The maximal projection volume conjecture for cross-polytopes

Let $\Diamond^n$ be the standard cross-polytope in $\mathbb{R}^n$, let $H_k$ be a $k$-dimensional subspace of $\mathbb{R}^n$, and let $\Diamond^n|H_k$ denote the orthogonal projection onto $H_k$. The $k$-dimensional cross-polytope has volume $2^k/k!$. Maximal projection volume conjecture. The projection satisfies

$$\mathop{\rm vol}(\Diamond^n|H_k)\leqslant\frac{2^k}{k!}.$$

The bound is attained only when $H_k$ is a coordinate subspace. The conjecture is known for hyperplanes, for two-dimensional projections, and for $k=3$ with $n\leqslant6$; the paper proves it for $k=2,3$, but the general case remains open.