Cube-section affine-cube extremality conjecture

Let $n>k\geq 1$, let $\Box^n=[-1,1]^n$, and let $H$ be a $k$-dimensional linear subspace. Write $C_{\Box}(n,k)2^k$ for the maximum volume among sections $\Box^n\cap H$ that are affine cubes. Cube-section affine-cube extremality conjecture. The maximal volume of a section of the cube $\Box^n$ by a $k$-dimensional linear subspace $H$ is attained on subspaces for which the section is an affine cube; equivalently,

$$\operatorname{vol}_{k}(\Box^n\cap H)\leq C_{\Box}(n,k)\operatorname{vol}_{k}\Box^k.$$

This conjecture gives the proposed sharp upper bound in the cases where Ball's general upper bounds are not known to be tight, namely when $k$ does not divide $n$ and $2k<n$.