The equal-edge projection conjecture for cube projections

Let $\operatorname{C}^n=[0,1]^n$ be the standard $n$-dimensional cube, and let $H$ be a subspace. Consider the orthogonal projection of $\operatorname{C}^n$ onto $H$. The edges of the cube are the coordinate-direction segments, whose projections are vectors in $H$.

Equal-edge projection conjecture. The maximum volume of a projection of $\operatorname{C}^n$ onto a subspace is attained when the projections of all edges of the cube have the same length.

Several extremal cases are known, including dimensions and codimensions specified in the source, and the conjecture is motivated by the fact that known maximizers in those cases have equal-length projected coordinate vectors. The general assertion remains open in the source.