The central-slice extremality conjecture for centrally symmetric polytopes

Let $P$ be a centrally symmetric $k$-dimensional polytope. Consider its $(k-1)$-dimensional slices, and let the upper bound mean the maximum possible number of vertices among such slices. A central slice is a slice through the center of $P$.

Central-slice extremality conjecture. For every centrally symmetric $k$-dimensional polytope $P$, the upper bound on the number of vertices of a $(k-1)$-dimensional slice of $P$ is attained by a central slice.

This is presented as a stronger statement that would imply the conjecture for cubes, because centrally symmetric polytopes can be realized as sections of cubes. The source gives no resolution beyond this implication and the reported computational evidence for the cube case.