The generic central slice conjecture for cubes

Let $C_d$ be the $d$-dimensional cube. For $k\leq d-1$, consider $k$-dimensional slices of $C_d$, and let the upper bound mean the maximum possible number of vertices among such slices. A central slice is one obtained by intersecting the cube with a linear subspace through its center, and it is generic when it avoids the relevant nongeneric incidences.

Generic central slice conjecture. The upper bound on the number of vertices of a $k$-dimensional slice of $C_d$ can be attained by a generic central $k$-dimensional slice for every $d$ and every $k\leq d-1$.

The preceding result establishes the analogous assertion for $(d-1)$-dimensional affine slices. The conjecture extends this phenomenon to all slice dimensions, and the source reports preliminary computational evidence.